A Blow-Up Criterion for the Nonhomogeneous Incompressible Navier--Stokes Equations
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GRE 杨鹏长难句 130 句(最简版只有句子)1.That sex ratio will be favored which maximizes the number of descendants an individual will have and hence the number of gene copies transmitted.2.,this is ,a desire to throw over reality a light that never was might give away abruptly to the desire on the part of what we might consider a novelist-scientist to record exactly and concretely the structure and texture of a flower .3.Hardy’s weakness derived from his apparent inability to control the comings and goings of these divergent impulses and from his unwillingness to cultivate and sustain the energetic and risky ones.4.Virginia Woolf’s provocative statement about her intentions in writing Mrs. Dalloway has regularly been ignored by the critics, since it highlights an aspect of her literary interests very different from the traditional picture of the "poetic" novelist concerned with examining states of reverie and vision and with following the intricate pathways of individual consciousness.5.As she put it in the common reader, “i t is safe to say that not a single law has beenframed or one stone set upon another because of anything Chaucer said or wrote; and yet, as weread him, we are absorbing morality at every pore.”6.With the conclusion of a burst activity , the lactic acid level is high in the body fluids , leaving the large animal vulnerable to attack until the acid is reconverted , via oxidative metabolism , by the liver into glucose , which is then sent (in part )back to the muscles for glycogen resynthesis .7.Although gutman admits that forced separation by sale was frequent, he shows that the slaves’ preference, revealed most clearly on plantations where sale was infrequent, was very much for stable monogamy.8.Gutman argues convincingly that the stability of the black family encouraged thetransmission of-and so was crucial in sustaining-the black heritage of folklore, music, and religious expression from one generation to another, a heritage that slaves were continually fashioning out of their african and american experiences.9.This preference for exogamy, gutman suggests, may have derived from west africanrules governing marriage, which, though they differed from one tribal group to another, all involved some kind of prohibition against unions with close kin.10.His thesis works relatively well when applied to discrimination against blacksin the united states, but his definition of racial prejudice as "racially-based negative prejudgments against a group generally accepted as a race in any given region of ethnic competition," can be interpreted as also including hostility toward such ethnic groups as the chinese in california and the jews in medieval europe.111.Such variations in size, shape, chemistry, conduction speed, excitation threshold, and the like as had been demonstrated in nerve cells remained negligible in significance for any possible correlation with the manifold dimensions of mental experience.12.It was possible to demonstrate by other methods refined structural differences among neuron types; however, proof was lacking that the quality of the impulse or its condition was influenced by these differences, which seemed instead to influence the developmental patterning of the neural circuits.13.Although qualitative variance among nerve energies was never rigidly disproved, the doctrine was generally abandoned in favor of the opposing view, namely, that nerve impulses are essentially homogeneous in quality and are transmitted as "common currency" throughout the nervous system.14.Other experiments revealed slight variations in the size, number, arrangement, and interconnection of the nerve cells, but as far as psycho neural correlations were concerned, the obvious similarities of these sensory fields to each other seemed much more remarkable than any of the minute differences.15.Although some experiments show that, as an object becomes familiar, its internal representation becomes more holistic and therecognition process correspondingly more parallel, the weight of evidence seems to support the serial hypothesis, at least for objects that are not notably simple and familiar.16.In large part as a consequence of the feminist movement, historianshave focused agreat deal of attention in recent years on determining more accurately thestatus of women in various periods.17.If one begins by examining why ancients refer to Amazons, it becomes clear that ancient Greek descriptions of such societies were meant not so much to represent observed historical fact - real amazonian societies - but rather to offer “moral lessons” on the supposed outcome of women’s rule in their own society.18.Thus, for instance, it may come as a shock to mathematicians to learn that the schrodinger equation for the hydrogen atom is not a literally correct description of this atom, but only an approximation to a somewhat more correct equation taking account of spin, magnetic dipole, and relativistic effects; and that this corrected equation is itself only an imperfect approximation to an infinite set of quantum field- theoretical equations.19.The physicist rightly dreads precise argument, since an argument that is convincingonly if it is precise loses all its force if the assumptions on which it is basedare slightlychanged, whereas an argument that is convincing though imprecise may well be stable under small perturbations of its underlying assumptions.220.However, as they gained cohesion, the bluestockings came to regard themselves as a women’s group and to possess a sense of female solidarity lacking inthe salonnieres,who remained isolated from one another by the primacy each held in her own salon.21.As my own studies have advanced, i have been increasingly impressed with the functional similarities between insect and vertebrate societies and less so with the structural differences that seem, at first glance, to constitute such an immense gulf between them.22.Although fiction assuredly springs from political circumstances, its authors react to those circumstances in ways other than ideological, and talking about novels and stories primarily as instruments of ideology circumvents much of the fictional enterprise.23.Is this a defect, or are the authors working out of, or trying to forge, adifferent kind of aesthetic,24.In addition, the style of some black novels, like jean toomer’s cane, verges on expressionism or surrealism; does this technique provide a counterpoint to the prevalent theme that portrays the fate, against which black heroes are pitted, a theme usually conveyed by more naturalistic modes of expression?25.Black fiction surveys a wide variety of novels, bringing to our attention in the process some fascinating and little-known works like Jjames Weldon Johnson’s Autobiography ofan Ex-Colored Man.26.Although these molecules allow radiation at visible wavelengths, where most of the energy of sunlight is concentrated, to pass through, they absorb some of the longer-wavelength, infrared emissions radiated from the earth’s surface, radiation that would otherwise be transmitted back into space.27.The role those anthropologists ascribe to evolution is not of dictating the details of human behavior but one of imposing constraints- ways of feeling, thinking, and acting that "come naturally" in archetypalsituations in any culture.28.Which of the following most probably provides an appropriate analogy from human mor phology for the “details” versus “constraints” distinction made in the passage in relation to human behavior?29. A low number of algal cells in the presence of a high number of grazers suggested, but did not prove, that the grazers had removed most of the algae.30.Perhaps the fact many of these first studies considered only algae of a size that could be collected in a net (net phytoplankton), a practice that overlooked the smaller 3phytoplankton (nannoplankton) that we now know grazers are most likely to feed on , led to a de-emphasis of the role of grazers in subsequent research.31.Studies by hargrave and geen estimated natural community grazing rates by measuring feeding rates of individual zooplankton species in the laboratory and then computing community grazing rates for field conditions using the known populationdensity of grazers.32.In the periods of peak zooplankton abundance, that is, in the late spring and in the summer, haney recorded maximum daily community grazing rates, for nutrient-poor lakes and bog lakes, respectively, of6.6 percent and 114 percent of daily phytoplankton production.33.The hydrologic cycle, a major topic in this science, is the complete cycle of phenomena through which water passes, beginning as atmospheric water vapor, passing into liquid and solid form as precipitation, thence along and into the ground surface, and finally again returning to the form of atmospheric water vapor by meansof evaporation and transpiration.34.Only when a system possesses natural or artificial boundaries that associate the water within it with the hydrologic cycle may the entire system properly be termed hydrogeologic.35.The historian Frederick J. Turner wrote in the 1890’s that the agrarian discontent that had been developing steadily in the unitedstates since about 1870 had been precipitated by the closing of the internal frontier - that is, the depletion of available new land needed for further expansion of the american farming system.36.In the early 1950’s, historians who studied preindustrial europe,whichwe may definehere as europe in the period from roughly 1300 to 1800, began, for the first timein largenumbers, to nvestigate more of the preindustrial european population than the 2 or 3 percent who comprised the political and social elite: the kings, generals, judges, nobles, bishops, and local magnates who had hitherto usually filled history books.37.Historians such as le roy ladurie have used the documents to extract case histories, which have illuminated the attitudes of different social groups ,theseattitudes include, butare not confined to, attitudes toward crime and the law,and have revealed howtheauthorities administered justice.38.It can be inferred from the passage that a historian who wished to compare crime rates per thousand in a european city in one decade of the fifteenth century with crime rates in another decade of that century would probably be most aided by better information about4which of the following?39.My point is that its central consciousness - its profound understanding of class and gender as shaping influences on people’s lives - owes much to that earlier literary heritage,a heritage that, in general, has not been sufficiently valued by most contemporary literary critics.40.Even the requirement that biomaterials processed from these materials be nontoxic to host tissue can be met by techniques derived from studying the reactions of tissue cultures to biomaterials or from short-term implants.41.But achieving necessary matches in physical properties across interfaces between living and nonliving matter requires knowledge of which molecules control the bonding of cells to each other - an area that we have not yet explored thoroughly.42.Islamic law is a phenomenon so different from all other forms of law - notwithstanding, of course, a considerable and inevitable number of coincidences with one or the otherof them as far as subject matter and positive enactments are concerned - thatits study is indispensable in order to appreciate adequately the full range ofpossible legalphenomena.43.,both jewish law and canon law are more uniform than islamic law.,though historicallythere is a discernible break between jewish law of the sovereign state of ancient israel and of the diaspora ,the dispersion of jewish people after the conquest of israel,, the spirit ofthe legal matter in later parts of the old testament is very close to that of the talmud, one of the primary codifications of jewish law in the diaspora.44.Islam, on the other hand, represented a radical breakaway from the arab paganism that preceded it; islamic law is the result of an examination, from a religious angle,(examination)of legal subject matter that was far from uniform, comprising as it did the various components of the laws of pre-islamic arabia and numerous legal elementstaken over from the non-arab peoples of the conquered territories.45.One such novel idea is that (idea) of inserting into the chromosomesof plantsdiscrete genes that are not a part of the plants’ naturalconstitution,specifically, the idea ofinserting into nonleguminous plants the genes, if they can be identified and isolated, that fit the leguminous plants to be hosts for nitrogen-fixing bacteria. Hence, (there is) the intensified research on legumes.46.It is one of nature’s great ironies that the availability of nitrogen in the soilfrequently sets an upper limit on plant growth even though the plants’ leavesare bathedin a sea of nitrogen gas. 547.Unless they succeed, the yield gains of the green revolution will be largely lost even if the genes in legumes that equip those plants to enter into a symbiosiswith nitrogenfixers are identified and isolated, and even if the transfer of those gene complexes,once they are found, becomes possible.48.Its subject,to use maynard mack’s categories,is "l ife-as- spectacle, " for readers,diverted by its various incidents, observe its hero odysseus primarily from without; the tragic iliad, however, presents "life-as- experience",readers are asked to identify with themind of achilles, whose motivations render him a not particularly likable hero.49.Most striking among the many asymmetries evident in an adult flatfish is eye placement: before maturity one eye migrates, so that in an adult flatfish both eyes are on the same side of the head.50. A critique of Handlin’s interpretation of why legal slavery did not appear until the 1660’s suggests that assumptions about the relation between slavery and racial prejudice should be reexamined, andthat explanations for the different treatment of black slaves in north and South America should be expanded.51.The best evidence for the layered-mantle thesis is the well- established fact that volcanic rocks found on oceanic islands, islands believed to result from mantle plumes arising from the lower mantle, are composed of material fundamentally different from that of the mid-ocean ridge system, whose source, most geologists contend, is the upper mantle.52.Some geologists, however, on the basis of observations concerningmantle xenoliths,argue that the mantle is not layered, but that heterogeneity is created by fluidsrich in "incompatible elements" ,elements tending toward liquid rather than solid state,percolating upward and transforming portions of the upper mantle irregularly,according to the vagaries of the fluids’ pathways.53.Fallois proposed that Proust had tried to begin a novel in 1908, abandoned it for what was to be a long demonstration of saint-beuve’s blindness to the real nature of greatwriting, found the essay giving rise to personal memories and fictional developments, and allowed these to take over it a steadily developing novel.54.The very richness and complexity of the meaningful relationships that keptpresenting and rearranging themselves on all levels, from abstract intelligence to profound dreamy feelings, made it difficult for proust to set them out coherently.55.But those of who hoped, with kolb, that kolb’s newly published complete edition ofProust’s correspondence for 1909 would document the process in greaterdetail are6disappointed.56.Now we must also examine the culture as we Mexican Americans have experienced it, passing from a sovereign people to compatriots with newly arriving settlers to, finally a conquered people a charter minority on our own land.57.It is possible to make specific complementarydna’s ,cdna’s,that can serve asmolecular probes to seek out the messenger rna’s ,mrna’s,of the peptide hormones. Ifbrain cells are making the hormones, the cells will contain these mrnas. if the products the brain cells make resemble the hormones but are not identical to them, then the cdna’sshould still bind to these mrna’s, but should not bind as tightly as they would to mrna’s for the true hormones.58.The molecular approach to detecting peptide hormones using cdna probes should also be much faster than the immunological method because it can take years of tedious purifications to isolate peptide hormones and then develop antiserums to them.59.Nevertheless, researchers of the pleistocene epoch have developed all sorts of more or less fanciful model schemes of how they would have arranged the ice age had they been in charge of events.60.This succession was based primarily on a series of deposits and events not directlyrelated to glacial and interglacial periods, rather than on the more usual modern method of studying biological remains found in interglacial beds themselves interstratified within glacial deposits.61.There have been attempts to explain these taboos in terms of inappropriate social relationships either between those who are involved and those who are not simultaneously involved in the satisfaction of a bodily need, or between those already satiated and those who appear to be shamelessly gorging.62.Man y critics of amily bronte’s novel wuthering heights see its second part as acounterpoint that comments on, if it does not reverse, the first part, where a "romantic" reading receives more confirmation.63.Granted that the presence of these elements need not argue an authorial awarenessof novelistic construction comparable to that of henry james, their presence does encourage attempts to unify the novel’s heterogeneous parts.64.This is not because such an interpretation necessarily stiffens into athesis,althoughrigidity in any interpretation of this or of any novel is always a danger,, but becausewuthering heights has recalcitrant elements of undeniable power that, ultimately, resist7inclusion in an all-encompassing interpretation.65.The isotopic composition of lead often varies from one source of common copper ore to another, with variations exceeding the measurement error; and preliminary studies indicate virtually uniform isotopic composition of the lead from a single copper-ore source.66.More probable is bird transport, either externally, by accidental attachmentof the seeds to feathers, or internally, by the swallowing of fruit and subsequent excretion ofthe seeds.67. A long-held view of the history of English colonies that became the United States has been that England’s policy toward these colonies before 1763 was dictated by commercial interests and that a change to a more imperial policy, dominated by expansionist militarist objectives, generated the tensions that ultimately led to the American Revolution.68.It is not known how rare this resemblance is, or whether it is most often seenin inclusions of silicates such as garnet, whose crystallography is generally somewhat similar to that of diamond; but when present, the resemblance is regarded as compelling evidence that the diamonds and inclusions are truly cogenetic.69.Even the "radical" critiques of this mainstream research model, such as the critique developed in divided society, attach the issue of ethnic assimilation too mechanically to factors of economic and social mobility and are thus unable to illuminate the cultural subordination of puerto ricans as a colonial minority.71.Open acknowledgement of the existence of women’s oppression was too radical for the united stated in the fifties, and beauvoir’s conclusion , that change in women’s economic condition, though insufficient by itself, “remains the basic factor ”in improvingwom en’s situation , was particularly unacceptable .72.Other theorists propose that the moon was ripped out of the earth’s rocky mantle bythe earth’s collision with another large celestial body after much of the earth’s iron fell to itscore.73.However, recent scholarship has strongly suggested that those aspects of early New England culture that seem to have been most distinctly puritan, such as the strong religious orientation and the communal impulse, were not even typical of New England as a whole, butwere largely confined to the two colonies of Massachusetts and Connecticut.74.Thus, what in contrast to the puritan colonies appears to davis to be peculiarly southern - acquisitiveness, a strong interest in politics and the law, and a tendency to cultivate metropolitan cultural models - was not only more typically english than the cultural patterns exhibited by puritan massachusetts and connecticut, but also almost certainly characteristic of most other early modern british colonies from barbados north 8to rhode island and new hampshire.75.Portrayals of the folk of Mecklenburg County, North Carolina, whom he remembers from early childhood, of the jazz musicians and tenement roofs of his Harlem days, of Pittsburgh steelworkers, and his reconstruction of classical Greek myths in the guise of the ancient black kingdom of Benin, attest to this.76. A very specialized feeding adaptation in zooplankton is that of the tadpolelike appendicularian who lives in a walnut-sized ,or smaller, balloon of mucus equipped with filters that capture and concentrate phytoplankton.77.These historians, however, have analyzed less fully the development of specifically feminist ideas and activities during the same period.78.Apparently most massive stars manage to lose sufficient material that their masses drop below the critical value of 1.4 m before they exhaust their nuclear fuel.79.This is so even though armed forces operate in an ethos of institutional change oriented toward occupational equality and under the federal sanction of equal pay for equal work.80.An impact (on the mars) capable of ejecting a fragment of the martian surface into an earth-intersecting orbit is even less probable than such an event on the moon, in view of the moon’s smaller size and closer proximity to earth.81.Not only are liver transplants never rejected, but they even induce a state of donor-specific unresponsiveness in which subsequent transplants of other organs,suchas skin, from that donor are accepted permanently.82.As rock interface are crossed, the elastic characteristics encountered ,by seismicwaves,generally change abruptly, which causes part of the energy to be reflected back to the surface, where it is recorded by seismic instruments.83.While the new doctrine seems almost certainly correct, the one papyrus fragment raises the specter that another may be unearthed, showing, for instance, thatit was aposthumous production of the danaid tetralogy which bested sophocles, and throwingthe date once more into utter confusion.84.The methods that a community devises to perpetuate itself come into being topreserve aspects of the cultural legacy that that community perceives as essential.985.Traditionally, pollination by wind has been viewed as a reproductive process marked by random events in which the vagaries of the wind are compensated for bythe generation of vast quantities of pollen, so that the ultimate production of new seeds is assured at the expense of producing much more pollen than is actually used.86.Because the potential hazards pollen grains are subject to as they are transported over long distances are enormous, wind pollinated plants have, in the view above, compensated for the ensuing loss of pollen through happenstance by virtue of producing an amount of pollen that is one to three orders of magnitude greater than the amount produced by species pollinated by insects.87.For example, the spiral arrangement of scale-bract complexes on ovule-bearing pinecones, where the female reproductive organs of conifers are located, is important to the production of airflow patterns that spiral over the cone’s surfaces, thereby passing airborne pollen from one scale to the next.88.Friedrich Engels, however, predicted that women would be liberated from the “social, legal, and economic subordination” of the family by technological developments that made possible the recruitment of “the whole female sex into public industry”.89.It was not the change in office technology, but rather the separation of secretarial work, previously seen as an apprenticeship for beginning managers, from administrative work that in the 1880’s created a new class of "dead-end" jobs, thenceforth considered "women’s work."90.The increase in the numbers of married women employed outside the home in the twentieth century had less to do with the mechanization of housework and an increase in leisure time for these women than it did with their own economic necessity and with high marriage rates that shrank the available pool of single women workers, previously, in many cases, the only women employers would hire.91.For one thing, no population can be driven entirely by density- independent factors all the time.92.In order to understand the nature of the ecologist’s investigation,we may think of thedensity-dependent effects on growth parameters as the "signal" ecologists aretrying to isolate and interpret, one that tends to make the population increase from relatively low values or decrease from relatively high ones, while the density-independent effects act to produce "noise" in the population dynamics.93.But the play’s complex view of black self-esteem and human solidarityas compatibleis no more "contradictory" than du bois’ famous, well-considered ideal of ethnic self-awareness coexisting with human unity, or fanon’s emphasis on an idealinternationalism that also accommodates national identities and roles.1094.In which of the following does the author of the passage reinforce his criticism of responses such as isaacs’ to raisin in the sun,95.Iinheritors of some of the viewpoints of early twentieth-century progressive historians such as beard and becker, these recent historians have put forward arguments that deserve evaluation.96.Despite these vague categories, one should not claim unequivocally that hostility between recognizable classes cannot be legitimately observed.97.Yet those who stress the achievement of a general consensus among the colonists cannot fully understand that consensus without understanding theconflicts that had to be overcome or repressed in order to reach it.98.It can be inferred from the passage that the author would be most likely to agree with which of the following statements regardingsocioeconomic class and support for the rebel and loyalist causes during the american revolutionary war,99.She wished to discard the traditional methods and established vocabularies of such dance forms as ballet and to explore the internal sources of human expressiveness.100.Although it has been possible to infer from the goods and services actually produced what manufactures and servicing trades thought their customers wanted, only a study of relevant personal documents written by actual consumers will provide a precise picture of who wanted what.101.Wwith regard to this last question, we might note in passing that thompson, while rightly restoring laboring people to the stage of eighteen-century english history, has probably exaggerated the opposition of these people to the inroads of capitalist consumerism in general: for example, laboring people in eighteen-century england readily shifted from home-brewed beer to standardized beer produced by huge, heavily capitalized urban breweries.102.The correlation of carbon dioxide with temperature, of course, does not establish whether changes in atmospheric composition caused the warming and cooling trends or were caused by their.103.Such philosophical concerns as the mind-body problem or, more generally, the nature of human knowledge they believe, are basic human questions whose tentative philosophical solutions have served as the。
小学上册英语第二单元测验卷(含答案)英语试题一、综合题(本题有100小题,每小题1分,共100分.每小题不选、错误,均不给分)1.What is the main ingredient in spaghetti?A. RiceB. FlourC. WheatD. Corn2.How many zeros are in one hundred?A. 1B. 2C. 3D. 4答案:B3.The _____ (种植技巧) improve garden productivity.4.My friend is a _____ (技术人员) who fixes computers.5.The chef, ______ (厨师), prepares wonderful dishes.6.My dad is a good ________ (厨师).7. Wall was built using forced ________ (劳动力). The Grea8.The __________ (科技应用) enhance everyday life.9.When it rains, I wear my ______ (雨衣) and jump in the ______ (水坑). It’s so much fun!10.What do you need to stay warm in winter?A. Ice creamB. JacketC. SandalsD. Shorts答案:B11.What is the capital of Malaysia?A. Kuala LumpurB. PenangC. Johor BahruD. Malacca12.What is the term for a young horse?A. FoalB. CalfC. KidD. Lamb答案:A13.I enjoy ______ (参加) art competitions.14.The chemical formula for ethanol is ________.15.The ________ (农业创新) drives progress.16.What is the largest land animal?A. LionB. ElephantC. GiraffeD. Rhino答案:B17. A reaction that produces heat is called an ______ reaction.18.I have a special ________ that reminds me of home.19.中国的历史上,各个________ (dynasties) 有着不同的特点与成就。
GRENO题难句摘录及分析GRE NO题难句摘录及分析1.That se_ ratio will be favored which ma_imizes the number ofdescendantsan individual will have and hence the number of gene copies transmitted.那种性别比例能在最大程度上增加一个个体所能拥有的后代数量,并因此能在最大程度上增加所传递到后代身上去的基因复制品的数量.2.Hardys weakness derived from his apparent inability to control thecomings and goings of these divergent impulses and from his unwillingness tocultivate and sustain the energetic and risky ones.哈代的缺陷一方面缘起于他的某种明显的无能,无法控制好那结不尽相同的创作冲动的穿梭往来;另一方面缘起于他不愿意去培养和维持那些富于生机活力和风险性强的创作冲动.3.Virginia Woolfs provocative statement about her intentions in writingMrs. Dalloway has regularly been ignored by the critics,since it highlights anaspect of her literary interests very different from the traditional picture ofthe 诗性小说家所形成的传统见解大相径庭.所谓的诗性小说家,所关注的是审视想入非非和白日梦幻的诸般状态,并致力于追寻个体意识的通幽曲径.4.Although Gutman admits that forced separation by sale was frequent,heshows that the slavespreference,revealed most clearly on plantations where salewas infrequent,was very much for stable monogamy.虽然古特曼承认,由于奴隶买卖而造成的被迫离散甚为频繁,但他还是证明,奴隶的偏爱在那些奴隶买卖并不频繁的种植园上被最为显著地揭示出来在很大程度上侧重于稳定的一夫一妻制.5.Gutman argues convincingly that the stability of the Black family encouraged the transmission of-and so was crucial in sustaining-the Blackheritage of folklore,music,and religious e_pression from one generation toanother,a heritage that slaves were continually fashioning out of their Africanandd American e_periences.古特曼人令人信服地论辨道,黑人家庭的稳定有助于包括民间传说.音乐.及宗教表达在内的黑人文化遗产一代一代传递下去,因而在维持文化遗产方面也起着至关重要的作用,而对于这种文化遗产,黑奴们不断地从其非洲和美洲的经历中予以丰富发展.6.This preference for e_ogamy,Gutman suggests,may have derived fromWestAfrican rules governing marriage,which,though they differed from one tribalgroup to another,all involved some kind of prohibition against unions with closekin.古特曼表示,这种对于外部通婚的偏爱很有可能缘起于西部非洲制约着婚姻的规定,尽管这些规定在一个和另一个部落群体之间不尽相同,但都涉及到某种对近亲联姻的禁止.8. Such variations in size,shape,chemistry,conduction speed,e_citationthreshold,and the like as had been demonstrated in nerve cells remainednegligible in significance for any possible correlation with the manifolddimensions of mental e_perience.9.Although qualitative variance among nerve energies was never rigidlydisproved,the trine was generally abandoned in favor of the opposingview,namely,that nerve impulses are essentially homogeneous in quality and aretransmitted as common currency throughout the nervous system. Othere_perimentsrevealed slight variations in the size,number,arrangement,and interconnection ofthe nerve cells,but as far as psychoneural correlations were concerned,theobvious similarities of these sensory fields to each other seemed much moreremarkable than any of the minute differences. Although some e_periments showthat,as an object becomes familiar,its internal representation becomes moreholistic and the recognition process correspondingly more parallel,the weight ofevidence seems to support the serial hypothesis,at least for objects that arenot notably simple and familiar.虽然某些实验表明,随着一个物体变得熟悉起来,其内心再现图像亦更具整体感,辨认过程相应地更趋于平行,但证据的砝码似乎在支持序列假设,至少是对于那些不甚简单.不甚熟悉的物体来说.10.In large part as a consequence of the feminist movement,historians havefocused a great deal of attention in recent years on determining more accuratelythe status of women in various periods.在很大程度上,由于女权主义运动的缘故,史学家近年来汇聚了大量的注意力,来更为准确地确定妇女在各个历史时期的地位.GRE NO题难句摘录及分析_.Thus,for instance,it may come as a shock to mathematicians to learn thatthe Schrodinger equation for the hydrogen atom is not a literally correctdescription of this atom,but only an appro_imation to a somewhat more correctequation taking account of spin,magnetic dipole,and relativistic effects;andthat this corrected equation is itself only an imperfect appro_imation to aninfinite set of quantum field- theoretical equations.因此,举例来说,对数学家而言,了解到下述情形可能会令其惊愕不已,即薛定谔的氢原子方程式并非是对该原子作出的一种绝然正确的描述,而仅仅是个近似值,趋近于一个在某种程度上更为正确的将自旋.磁性偶极子.以及相对论效应考虑在内的方程式;而这个得以纠正的方程式就其本身而言也只是一个不完美的近似值,趋近于无穷无尽的一整套量子场论方程式._.The physicist rightly dreads precise argument,since an argument that isconvincing only if it is precise loses all its force if the assumptions on whichit is based are slightly changed,whereas an argument that is convincing thoughimprecise may well be stable under small perturbations of its underlyingassumptions.物理学家恐惧于那些精确无误的论据不无道理,因为某种只有在它是精确无误的条件下才令人置信的论据,一旦它赖于建立其上的假设稍有变化,便会失去它一部的作用;而与此相反,一个尽管并不精确无误但却令人置信的论据,在其基本假设稍微受干扰的情况下,仍然有可能是站得住脚的._.However,as they gained cohesion,the Bluestockings came to regardthemselves as a womens group and to possess a sense of female solidarity lackingin the salonnieres,who remained isolated from one another by the primacy eachheld in her own salon.起初,蓝袜女们确实模仿了法国沙龙女主人,将男性襄括到其小圈子中来.然则,随着她们获得的凝聚力,她们渐趋将自己视作一女性团体,并拥有了一种妇女团结意识,而这种意识在法国沙龙女主人身上则荡然无存,因为她们每个人在其自己的沙龙中自视甚高而彼此孤立隔绝开来._.As my own studies have advanced,I have been increasingly impressed withthe functional similarities between insect and vertebrate societies and less sowith the structural differences that seem,at first glance,to constitute such animmense gulf between them.随着我的研究不断深入,我对昆虫和脊椎动物群落之间的功能类似性印象愈来愈深刻,而对结构上的差异印象愈发淡漠,虽然这些结构上的差异初看上去似乎构成了二者间一条无法愈越的鸿沟._. Although fiction assuredly springs from political circumstances,itsauthors react to those circumstances in ways other than ideological,and talkingabout novels and stories primarily as instruments of ideology circumvents muchof the fictional enterprise.虽然小说无疑起源于政治情状,但其作者则是以非意识形态的方式对这些政治情状作出反应的,而将小说和故事主要地当作意识形态的工具来探讨,会在相当程度上阻碍小说事业.GRE NO题难句摘录及分析_.Is this a defect,or are the authors working out of,or trying to forge,adifferent kind of aesthetic?这究竟是一种缺陷呢,或者是否表明,这些作者是在按照一种与众不同的美学体系进行创作,抑或是在试图创立一种与众不同的美学体系?_.Black Fiction surveys a wide variety of novels,bringing to our attentionin the process some fascinating and little-known works like James WeldonJohnsons Autobiography of an E_- Colored Man.《黑人小说》考察了极为广泛的一系列小说,在此过程中让我们注意到了某些引人入胜但却鲜为人知的作品,如詹姆斯.韦尔登.约翰逊的《一个曾经是有色人的自传》._.Although these molecules allow radiation at visible wavelengths,wheremost of the energy of sunlight is concentrated,to pass through,they absorb someof the longer-wavelength,infrared emissions radiated from the Earthssurface,radiation that would otherwise be transmitted back into space.虽然这些分子允许可见波长的辐射自然表露的情感.思维.以及行动方式.20.A low number of algal cells in the presence of a high number of grazerssuggested,but did not prove,that the grazers had removed most of the algae.在存在大量食草动物的同时却只有少量的水藻花粉囊,这暗示出但没能证明食草动物已吞噬了大部分水藻._.Studies by Hargrave and Geen estimated natural community graz-ing ratesby measuring feeding rates of individual zooplankton species in the laboratoryand then computing community grazing rates for field conditions using theknownpopulation density of grazers.由哈格雷夫和吉恩所进行的研究,对自然条件下的群落食草比例进行了估计,其手段是通过测量出实验室内单独的浮游动物种类的结食比例,然后利用已知的食草动物种群密度,计算出实地状况下的群落食草比例._.In the periods of peak zooplankton abundance,that is,in the late springand in the summer,Haney recorded ma_imum daily com-munity grazing rates,fornutrient-poor lakes and bog lakes,respectively,of 6.6 percent and _4 percent ofdaily phytoplankton production.在浮游动物数量激增的高峰期,亦即在春季后期以及夏季,哈尼记录了最大程度上的每日群落食草比率,对于营养物不充足的湖和沼泽湖而言,分别为每日浮游植物繁殖量的6.6%和_4%.23.The hydrologic cycle,a major topic in this science,is the complete cycleof phenomena through which water passes,beginning as atmospheric watervapor,passing into liquid and solid form as precipitation,thence along and intothe ground surface,and finally again returning to the form of atmospheric watervapor by means of evaporation and transpiration.水文循环,作为该学科中的一个主要课题,指的是水所经过的诸现象的整个循环过程,开始时是作为大气中的水蒸气,转而作为雨.雪.露.雹一类的降水量经过液体和固体形态,由此而沿着地层表面分布或进入地层表面,最终通过蒸发和散发作用再度回复到大气水蒸气的形态.24.The historian Frederick J. Turner wrote in the _90s that the agrariandiscontent that had been developing steadily in the United States since about_70 had been precipitated by the closing of the internalfrontiers,historianswho studied preindustrial Europe began,for the first time in large numbers,toinvestigate more of the preindustrial European population than the 2or3 percentwho comprised the political and social elite:thekings,generals,judges,nobles,bishops,and local magnates who had hitherto usuallyfilled history books.二十世纪五十年代早期,研究前工业化时代欧洲的史学家,首次以众多的人数,开始调查前工业化时代欧洲人口中的大多数,而非那些构成了政治与社会精英阶层的百分之二或三的人口,即国王.将军.法官.贵族.主教.以及地方上的达官显贵,而正是这部分人一直到那时为止普遍充斥于史学著作.GRE NO题难句摘录及分析26.Historians such as Le Roy Ladurie have used the uments to e_tractcase histories,which have illuminated the attitudes of different social groupsand have revealed how the authorities administered justice.象勒罗伊·拉迪里一类的史学家利用这些文献史料从中挖掘出某些个案史来,阐明了不同社会群体的态度,并揭示出当局是如何执行审判的.27.My point is that its central consciousnesss livesowes much to thatearlier literary heritage,a heritage that,in general,has not been sufficientlyvalued by most contemporary literary critics.我的论点是,其作品的中心意识它将阶级和性别作为人们生活的决定性影响而作出的深邃理解在很大程度上借鉴了那个早期的文学遗产,而这一遗产就总体而言还尚未获得大多数当代文学评论家的足够重视.28.But achieving necessary matches in physical properties across interfacesbetween living and nonliving matter requires knowledge of which moleculescontrol the bonding of cells to each othernaturalconstitution:specifically,theidea of inserting into nonleguminous plants the genes,if they can be identifiedand isolated,that fit the leguminous plants to be hosts for nitrogen-fi_ingbacteria. Hence,the intensified research on legumes.这其中的一个新颖思想就是,在植物的染色体内注入并非是该植物自然构造一个部分的那些不相关联的因基:具体而言,这一思想是,在非豆科植物内注入这样一些基因,倘若这些基因可被辨识出来并被分离开来,而这些基因业已使豆科植物宜于充当那些具备固氮作用的细菌的寄主.由此,对豆科植物的研究日趋深入.33.It is one of natures great ironies that the availability of nitrogen inthe soil frequently sets an upper limit on plant growth even though theplantsleaves are bathed in a sea of nitrogen gas.下述情形真可谓是自然界的一个莫大讽刺:土壤中所能获得的氮肥量往往对植物的生长构成了一个上限,虽然植物的叶子被沐浴在一片氮气的海洋中.34.Unless they succeed,the yield gains of the Green Revolution will belargely lost even if the genes in legumes that equip those plants to enter intoa symbiosis with nitrogen fi_ers are identified and isolated,and even if thetransfer of those gene comple_es,once they are found,becomes possible.除非他们能取得成功,不然的话,绿色革命的产量收益将在很大程度上损失殆尽,即使豆科植物中使这些植物有条件进入到与固氮细菌共生关系的基因可被辨识出来和分离开来的话,且即使这些基因综合体,一旦被发现之后,其移植得以成为可能的话.GRE NO题难句摘录及分析。
2012年中国社会科学院考博英语真题试卷(题后含答案及解析)题型有:1. Structure and V ocabulary 2. Grammar 3. Reading Comprehension 4. English-Chinese Translation 5. Chinese-English TranslationStructure and V ocabulary1.But two hurdles stand in the way of Russia’s realizing its space dreams: a collapsing public-education system and a brain drain that for decades has been siphoning off the country’s highly trained engineers as they move to better-paying jobs in the West.A.obstaclesB.propheciesC.hasslesD.outcomes正确答案:A解析:A项意为“障碍”;B项意为“预言”;C项意为“激战”;D项意为“结果”。
句子中画线单词hurdles意为“篱笆,障碍”,因此,A选项符合题意。
2.Its subject is “life-as-spectacle”, for readers, diverted by its various incidents, observe its hero Odysseus primarily from without: the tragic Iliad, however, presents “life-as-experience”: readers are, asked to identify with the mind of Achilles, whose motivations render him a not particularly likeable hero.A.insideB.outsideC.lackingD.surrounding正确答案:B解析:A项意为“内部”;B项意为“外部,外界”;C项意为”缺乏”;D 项意为“周围的,附近的”。
小学下册英语第六单元测验卷(有答案)英语试题一、综合题(本题有50小题,每小题1分,共100分.每小题不选、错误,均不给分)1 The _______ (狗) loves to bark at mailmen.2 The body part that helps us breathe is called the ______.3 将图片对应的单词涂色,并将单词抄写在四线三格内。
4 The element with the chemical symbol Na is ______.5 I have a brown ___. (dog)6 The capital of Iceland is __________.7 The ________ was a defining battle in the campaign for freedom.8 The chemical formula for -pentanol is ______.9 The _____ (caterpillar) is on the leaf.10 The __________ (历史的启示) inspires action.11 A __________ is a substance that increases the rate of a reaction without being consumed.12 My dad inspires me to be __________ (勇敢的) in life.13 The three states of matter are solid, liquid, and _____.14 The teacher, ______ (老师), encourages creativity in class.15 A __________ is a narrow strip of land connecting two larger land areas. (地峡)16 What is the most popular sport in the world?A. BasketballB. SoccerC. BaseballD. Tennis17 What is the color of grass?A. BlueB. YellowC. GreenD. Red18 We have a _____ (邀请) for the wedding.19 What do you call a small, furry animal that loves to dig?A. HamsterB. RabbitC. MoleD. Squirrel答案:C20 What is the color of the sky on a clear day?A. GreenB. BlueC. OrangeD. Purple21 The _______ (青蛙) is green and brown.22 h Revolution began in __________ (1789) and led to the rise of democracy. The Fren23 The cat is ________ by the window.24 How many colors are in a rainbow?A. 5B. 6C. 7D. 825 The _____ (灌木) provide shelter for small animals.26 What do we call a type of poetry that tells a story?A. LyricB. EpicC. OdeD. Sonnet27 What is the term for a young seal?a. Pupb. Calfc. Kitd. Cub答案:a28 I saw a _______ (小鹿) grazing peacefully.29 The __________ can show evidence of past tectonic activity.30 The Amazon River is located in _______.31 What is the tallest mountain in the world?A. K2B. KilimanjaroC. Mount EverestD. Mount Fuji答案:C32 What do you use to measure length?A. ScaleB. RulerC. ClockD. Thermometer33 What sport uses a bat and ball and involves running bases?A. BasketballB. TennisC. BaseballD. Soccer答案:C34 The ________ was a famous rebellion against British rule in India.35 The mountains are _____ (tall/short) and beautiful.36 My mom loves to create ____ (recipes) for the family.37 The __________ can be classified based on their appearance.38 What is the capital of Egypt?A. CairoB. AlexandriaC. LuxorD. Giza39 What is the capital of the Bahamas?A. NassauB. FreeportC. EleutheraD. Long Island答案: A. Nassau40 We are learning about ______ (space) in class.41 The ____ lounges in the sun and enjoys the warmth.42 The engineer, ______ (工程师), works on building projects.43 The _____ (小羊) frolics in the sunny meadow. 小羊在阳光明媚的草地上嬉戏。
The Integrity of Human and Nature———An Analysis of Gone with the Wind from thePerspective of Eco-feminismIt’s generally accepted that Gone with the Wind is a popular novel except a few people thought it is not worth reading. Due to some controversial problems had been discussed in the book so the novel was neglected for a long time by critics, such as its descriptions of the black, of the reconstruction after the war and of ku klux klan and so on. However, Donald W. Miller praised the book as a great epic which influences a number of people of different generations and nations. Recently people come to realize there is something significant behind the big work so that more and more people are addressing themselves to study it from many perspectives.Firstly, there are some essays about the relationship between Margaret Mitchell and Scarlett. Carolyn Gage, described a true-life Mitchell who suffered great grief after her fiancé’s death and then got flash marriage to a bootlegger which took her little happiness, later, even though she enjoyed a fast pace of life she could not escape the stigma of her gender, so when Mitchell reinvested herself in the novel she rewrote life the way she thought it should have been, therefore, she created a right man instead of the dashing and sexually charismatic alcoholic. In this way, Gage concluded that Mitchell’s writing purpose was to reveal thedelusion of female life and escape from her shadowy preceding life, so Gage appealed that females should not live in dream under the patriarchal society, but take Scarlett for example, to live in present and pursue what they need.Secondly, female consciousness of tomorrow catches people’s attention. Luo Minyu considered Scarlett’s consciousness of tomorrow as a kind of expectation: expecting the equality between men and women, a peaceful world and happy homeland and a true love.Thirdly, another popular topic is the comparison between Scarlett and some females in other works, such as Jane Eyre or Wang Xifeng. Zhai Xu compared Scarlett and Wang Xifeng and thought that they were a typical paragon of feminism who possessed the awareness of Women’s Awakening; both were brave, strong-minded, independent and rebellious against the restrictions which set against the females by the society of their own time. Fourth, the Tara Theme is also noticeable, and Vicki L. Eaklor pointed out that “Tara Theme” was the story’s musical leitmotif as Tara was repeated frequently in the novel and it was Scarlette’s spiritual pillar. The war destroys Scarlett’s family as well as her happy life, but it did not defeat Scarlett as she had Tara beside her which gave her strength, every time she got into trouble she turned to Tara for comfort and encouragement, while Tara had trouble,Scarlett safeguarded it by all means.Besides, a majority of professors analyzed the novel from the perspective of feminism. Both Gu Shaoyang and Wang Yanli acclaimed the novel was a great one, not only because it reflected female’s self-awareness, especially after the war broken out and women had to go out to realize their economic independence, but also because it confirmed women’s contribution to the society, praised their fraternity, selflessness and rebellious spirit, and revealed the belief that women should be liberated from families. Wu Shijuan and Xie Jingzhi also thought that Scarlett was a heroic female since she dared to break the traditional prejudice that “females are inferior to males”. She was brave enough to walk out of the house and entered into the “patriarchy’s world”and competed against the men for her economic independence. In Liu Xiaoyong’s essay, he pointed out that Scarlett was increasingly maturing and in the end became a Southern new woman with strong feminist leanings along with the advance of war and industrialization. When it comes to Zhou Yaming, he deeply analyzed Scarlett’s struggle, success and failure and her psychical conflict in her pursuit of material and love, and then he put forward that Rhett, one of the representatives of male, his leaving possibly meant the male’s loss to conquer female; meanwhile, Scarlett suffered great loss: she lost her true love and happy life. And in the game no one won. Therefore, Zhou concluded that this complex bisexual relationship reflected Mitchell’s desire, rebel, conquer andregression toward male, and her wish of a harmony bisexual relationship.In addition, some critics tried to analyze the novel from the perspective of eco-feminism, which studies the close relationship between female and nature. This is a most novel and interesting way to analyze the book. Zhen Jigen generalized that Tara and the lumber mill were the resources for Scarlett’s eco-feminism, in which it was not the male who conquered and oppressed the land as well as conquered the female, but a young lady who called Scarlett conquered the land and men’s will, as Scarlett unexpectedly disobeyed her second husband’s will to operate the lumber mill and proved rather successfully; when she came back to Tara during the war she could manage the plantation without the help of man and finally possessed the supreme dominion. So Zhen concluded that these behaviors were a good match with the ecofeminist’s opinion, that female and nature were linked together in the term of creating and maintaining life, and natural process abode by female principles of motile creativity, diversity, integrity, sustainability and sanctity of life. When it came to Cao Peihong, she focused on the three kinds of links between nature and women. First link was a symbolic one which stated women’s “naturalized”and nature’s “feminized”; second one was the experiential link about the hardships and humiliation suffered by women and nature in patriarchal societies; third one was about females and nature’status link which embodied in females’folly and nature’identification as the field and background of human activities. Then she discussed in detail about their mutual dependence ——females were caretakers of nature while nature was the haven for females. Take house (which was an important part of nature according to eco-feminist) for example, Cao acclaimed women playing a great role in managing the house and plantation before, during and after the war; and the reasons why women were willing to commit any kind of sacrifice lied in houses’illustration of women’s beauty and traits, as houses were the place for women to convey their feeling through the order and decorations and furniture; but the most important thing was women had a deep love for those dwellings, in a word, houses as well as nature were haven for women in hard time.To sum up, the above critical reviews on Gone with the Wind may represent the major achievements of the studies of this novel in the recent decades. These studies have effectively advanced the exploration of the novel, broadened readers’vision as well as deepened readers’comprehension of it. The eco-feminism theory sums up early scholars’two separate topic of feminism and Tara Theme, and then analyzes the novel from a new way which links nature and female together. However, the essays of eco-feminism only use the theory to analyze Scarlett’s eco-feminism thoughts and her relative manifestations, in the novel it appears as Scarlett’s deep attachment to Tara and her every endeavor toprotect it from being destroyed, in turn, Tara is her spiritual pillar which will help her come through every difficulty. But the purpose and significance of the eco-feminism have not been further exposited. On the one hand, the reason why Scarlett tried her best to strive for more rights and final say was not only to satisfy her vanity but to be as free as men and to be treated equally as men. On the other hand, the purpose of eco-feminism is not to establish a society where female are at last over or conquer male, but pursue the balance between female and male, human and nonhuman nature, especially nowadays, due to many reasons, women haven’t fully displayed their talents to better the world which suffers the crises of ecological environment and threat of wars, which shows the relationship between nature and human beings is not so harmonious.Therefore, in this thesis, the eco-feministic point of view will be used to figure out the close relationship between female and nature in the novel to encourage the society especially the patriarchy-centered society to treat feminism seriously and respect them; to encourage the feminists to care for the ecology, and to do more by their strength to be against wars in the world since wars destroy the lands and kill people which both belong to the nature;to encourage the society to build a new environment where female and male, human and nonhuman nature are equal and harmonious.Bibliography:[1]Cao Peihong. A Study of Eco-feminism in Gone with the wind[D].Northeast Normal University, 2007.[2]Gage, Carolyn.Tara and Other Lies-Margaret and the real RhettButler[J]. On the Issues, V ol. 6, No. 2; Pg. 34; 1997.[3]Liu Xiaoyong.Scarlett O Hara: An Increasingly Maturing SouthernNew Woman [D]. Hunan Normal University, 2003.[4]Madsen, L. Debirag. Feminist Theory and Literary Practice[M]. PlutoPress, 2006.[5]Miller, W. Donald. “Gone With the Wind: An American Epic.”/miller/miller22.html.2007-04/2008-05-23.[6]Vicki L. Eaklor. Striking Chords and Touching Nerves: Myth andGender in Gone with the wind./2002/features/gwtw/[7]Zhou Xianghua.The Fate Concerto of Nature and Women [D].Nanchang Univeristy, 2007.[8]顾韶阳,王丽艳. 《飘》与妇女觉醒——《飘》中女性价值观浅析[J]. 西安外国语学院报, 2003(09): 67-69.[9]罗闵钰. 解读《飘》的“明天意识”[J]. 昆明大学学报,2005(1):38-42.[10]吴世娟. 漫谈《飘》中的女性主义思想[J]. 焦作大学学报,2006(10): 28-29.[11]谢景芝. 《飘》中郝思嘉形象魅力解读[J]. 河南大学学报,2005(2):85-88.[12]翟旭.《红楼梦》中的王熙凤与《飘》中的郝思嘉[J]. 沈阳师范大学学报, 2007(5):113-115.[13]张京晨.《飘》中的战争与女性[J]. 北京市计划劳动管理干部学院学报,2006(1): 61-63.[14]郑际根. 种植园土壤上的女权主义之花——《飘》中女主人公郝思嘉[J]. 湘潭师范学院学报,2007(06): 118-119.[15]周亚明. 郝思嘉女性意识的深度审视[J]. 洛阳师范大学学院学报,2006(06): 66-68.。
三年级基础能源英语阅读理解25题1<背景文章>Solar energy is a very important kind of energy. It comes from the sun. The sun gives out a great deal of energy every day. We can use solar energy in many ways. For example, we can use solar panels to collect the sun's energy. Then we can turn it into electricity.Solar energy has many advantages. First, it is clean. It doesn't produce any pollution like some other kinds of energy. Second, it is renewable. As long as there is a sun, we can always get solar energy. In our daily life, we can see solar energy used in many places. Some houses use solar energy to heat water. And some street lights are powered by solar energy too.1. Where does solar energy come from?A. The moon.B. The sun.C. The earth.D. The stars.答案:B。
解析:文章明确提到Solar energy is a very important kind of energy. It comes from the sun.,所以太阳能来自太阳,答案为B。
SIAM J.M ATH.A NAL.c 2006Society for Industrial and Applied Mathematics Vol.37,No.5,pp.1417–1434A BLOW-UP CRITERION FOR THE NONHOMOGENEOUSINCOMPRESSIBLE NA VIER–STOKES EQUATIONS∗HYUNSEOK KIM†Abstract.Let(ρ,u)be a strong or smooth solution of the nonhomogeneous incompressible Navier–Stokes equations in(0,T∗)×Ω,where T∗is afinite positive time andΩis a bounded domain in R3with smooth boundary or the whole space R3.We show that if(ρ,u)blows up at T∗,thenT∗0|u(t)|sL r w(Ω)dt=∞for any(r,s)with2s+3r=1and3<r≤∞.As immediate applications,we obtain a regularity theorem and a global existence theorem for strong solutions.Key words.blow-up criterion,nonhomogeneous incompressible Navier–Stokes equations AMS subject classifications.35Q30,76D05DOI.10.1137/S00361410044421971.Introduction.The motion of a nonhomogeneous incompressible viscousfluid in a domainΩof R3is governed by the Navier–Stokes equations(ρu)t+div(ρu⊗u)−Δu+∇p=ρf,(1.1)ρt+div(ρu)=0in(0,∞)×Ω,(1.2)div u=0(1.3)and the initial and boundary conditions(ρ,ρu)|t=0=(ρ0,ρ0u0)inΩ,u=0on(0,T)×∂Ω,(1.4)ρ(t,x)→0,u(t,x)→0as|x|→∞,(t,x)∈(0,T)×Ω.Here we denote byρ,u,and p the unknown density,velocity,and pressurefields of thefluid,respectively.f is a given external force driving the motion.Ωis either a bounded domain in R3with smooth boundary or the whole space R3.Throughout this paper,we adopt the following simplified notation for standard homogeneous and inhomogeneous Sobolev spaces:L r=L r(Ω),D k,r={v∈L1loc(Ω):|v|D k,r<∞};H k,r=L r∩D k,r,D k=D k,2,H k=H k,2;D10={v∈L6:|v|D1<∞and v=0on∂Ω};H10=L2∩D10,,D10,σ={v∈D10:div v=0inΩ};|v|D k,r=|∇k v|L r and|v|D10=|v|D10,σ=|∇v|L2.Note that the space D10is the completion of C∞c(Ω)in D1,and thus there holds the following Sobolev inequality:|v|L6≤2√3|v|D1for all v∈D10.(1.5)∗Received by the editors March17,2004;accepted for publication(in revised form)May9,2005; published electronically January10,2006.This work was supported by the Japan Society for the Promotion of Science under the JSPS Postdoctoral Fellowship for Foreign Researchers./journals/sima/37-5/44219.html†School of Mathematics,Korea Institute for Advanced Study207-43Cheongnyangni2-dong, Dongdaemun-gu,Seoul,130-722,Korea(khs319@postech.ac.kr,khs319@kias.re.kr).14171418HYUNSEOK KIMFor a proof of (1.5),see sections II.5and II.6in the book by Galdi [11].The global existence of weak solutions has been established by Antontsev and Kazhikov [1],Fernandez-Cara and Guillen [10],Kazhikov [13],Lions [21],and Simon [26,27].From these results (see [21]in particular),it follows that for any data (ρ0,u 0,f )with the regularity0≤ρ0∈L 32∩L ∞,u 0∈L 6,and f ∈L 2(0,∞;L 2),there exists at least one weak solution (ρ,u )to the initial boundary value problem (1.1)–(1.4)satisfying the regularity ρ∈L ∞(0,∞;L 32∩L ∞),√ρu ∈L ∞(0,∞;L 2),and u ∈L 2(0,∞;D 10,σ)(1.6)as well as the natural energy inequality.Then an associated pressure p is determinedas a distribution in (0,∞)×Ω.But the global existence of strong or smooth solutions is still an open problem and only local existence results have been obtained for sufficiently regular data satisfying some compatibility conditions.For details,we refer to the papers by Choe and the author [6],Kim [15],Ladyzhenskaya and Solonnikov [19],Okamoto [22],Padula [23],and Salvi [24].In particular,it is shown in [6](see also [7])that if the data ρ0,u 0,and f satisfy 0≤ρ0∈L 32∩H 2,u 0∈D 10,σ∩D 2,−Δu 0+∇p 0=ρ120g,(1.7)f ∈L 2(0,∞;H 1),and f t ∈L 2(0,∞;L 2)for some (p 0,g )∈D 1×L 2,then there exist a positive time T and a unique strongsolution (ρ,u )to the problem (1.1)–(1.4)such that ρ∈C ([0,T ];L 32∩H 2),u ∈C ([0,T ];D 10,σ∩D 2)∩L 2(0,T ;D 3),(1.8)u t ∈L 2(0,T ;D 10,σ),and √ρu t ∈L ∞(0,T ;L 2).Moreover,the existence of a pressure p in C ([0,T ];D 1)∩L 2(0,T ;D 2)can be deducedfrom (1.1)–(1.3).See [5]for a detailed argument.Let (ρ,u )be a global weak solution to the problem (1.1)–(1.4)with the data (ρ0,u 0,f )satisfying condition (1.7).Then from the above local existence result and weak-strong uniqueness results in [6]and [21],we conclude that the solution (ρ,u )satisfies the regularity (1.8)for some positive time T .One fundamental problem in mathematical fluid mechanics is to determine whether or not (ρ,u )satisfies (1.8)for all time T .As an equivalent formulation,we may ask the following.Fundamental question 1.1.Does the solution (ρ,u )blow up at some finite time T ∗?Such a time T ∗,if it exists,is called the finite blow-up time of the solution (ρ,u )in the class H 2.In spite of great efforts since the pioneering works by Leray [20]in 1930s,there have been no definite answers to the fundamental question even for the case of the homogenous Navier–Stokes equations with only some blow-up criteria available.The first criterion is due to Leray [20]who proved,among other things,that if T ∗is the finite blow-up time of a strong solution u to the Cauchy problem for the homogeneous Navier–Stokes equations,then for each r with 3<r ≤∞,there exists a constant C =C (r )>0such that |u (t )|L r ≥C (T ∗−t )−12(1−3r )for all near t <T ∗.(1.9)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1419This estimate near the blow-up time was extended by Giga [12]to the case of bounded domains.An immediate consequence of (1.9)is the following well-known blow-up criterion in terms of the so-called Serrin class (see [9,25,29]):T ∗0|u (t )|s L r dt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.10)By virtue of Sobolev inequality (1.5),we deduce from (1.10)thatT ∗|∇u (t )|4L 2dt =∞.(1.11)Further generalizations of (1.10)and (1.11)have been obtained by Beirao da Veiga [2],Berselli [3],Chae and Choe [4],and Kozono and Taniuchi [17].The major purpose of this paper is to prove the blow-up criterion (1.10)for strong solutions of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3).In fact,we establish a more general result.To state our main result precisely,we first introduce the notion of the blow-up time of solutions in the class H 2m with m ≥1.Let (ρ,u )be a strong solution to the initial boundary value problem (1.1)–(1.4)with the regularityρ∈C ([0,T ];L 32∩H 2m ),u ∈C ([0,T ];D 10,σ∩D2m)∩L 2(0,T ;D 2m +1),∂j t u ∈C ([0,T ];D 10,σ∩D2m −2j)∩L 2(0,T ;D 2m −2j +1)for 1≤j <m,(1.12)∂m t u ∈L 2(0,T ;D 10,σ),and √ρ∂m t u ∈L ∞(0,T ;L 2)for any T <T ∗,where T ∗is a finite positive time.Then we can defineΦm (T )=1+sup 0≤t ≤T|∇ρ(t )|H 2m −1+|u (t )|D 10∩D 2m + T|u (t )|2D 2m +1dt+sup1≤j<msup 0≤t ≤T|∂jt u (t )|D 10∩D 2m −2j +T|∂jt u (t )|2D 2m −2j +1dt(1.13)+ess sup 0≤t ≤T|√ρ∂mtu (t )|L 2+T|∂mt u (t )|2D 10dtfor any T <T ∗.Hereafter we use the obvious notation|·|X ∩Y =|·|X +|·|Yfor (semi-)normed spaces X,Y.Definition 1.2.A finite positive number T ∗is called the finite blow-up time ofthe solution (ρ,u )in the class H 2m ,provided thatΦm (T )<∞for0<T <T ∗andlim T →T ∗Φm (T )=∞.We are now ready to state the main result of this paper.Theorem 1.3.For a given integer m ≥1,we assume that∂mtf ∈L 2(0,∞;L 2)and∂jt f ∈L 2(0,∞;H 2m −2j −1)for 0≤j <m.Let (ρ,u )be a strong solution of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3)satisfying the regularity (1.12)for any T <T ∗.If T ∗is the finite blow-up time of (ρ,u )in the class H 2m ,then we haveT ∗|u (t )|s L r wdt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.14)1420HYUNSEOK KIMHere L r w denotes the weak L r-space,that is,the space consisting of all vector fields v ∈L 1loc (Ω)such that |v |L r w=sup α>0α|{x ∈Ω:|v (x )|>α}|1r <∞for 3<r <∞and |v |L ∞w=|v |L ∞<∞.In the case when 3<r <∞,L ris a proper subspace of L r w (|x |−3/r ∈L r w (R 3)for instance)and so Theorem 1.3is in fact a generalizationof the blow-up criterion (1.10)due to Leray and Giga even for the homogeneous Navier–Stokes equations.Theorem 1.3is proved in the next two sections.In section 2,we provide a proof of the theorem for the very special case m =1.Our method of the proof is quite well known in the case of the homogeneous Navier–Stokes equations and was also applied in an earlier paper [6]by Choe and the author to the nonhomogeneous case:combining classical regularity results on the Stokes equations with H¨o lder and Sobolev inequal-ities,we show that Φ1(T )is bounded in a double exponential way by T0|u (t )|s L r wdt for any T less than the blow-up time T ∗.But the use of weak Lebesgue spaces in space variables makes it more difficult to estimate the nonlinear convection term.To overcome this technical difficulty,we utilize some basic theory of the Lorenz spaces developed in [18]and [30].See the derivations of (2.6)and (2.7)for details.Concern-ing the proof for the general case m ≥2,the basic idea is also to show that Φm (T )isbounded in some specific way by T0|u (t )|s L r wdt for any T <T ∗.Such an approach is more or less standard in the case of the homogeneous Navier–Stokes equations,but its extension to the nonhomogeneous case is not straightforward and indeed much complicated due to the evolution of the density.A detailed argument is provided in section 3.Some corollaries of Theorem 1.3can be deduced from a local existence result on strong solutions in the class H 2m .For instance,as an immediate consequence of Theorem 1.3,the local existence result in the class H 2,and the weak-strong uniqueness result,we obtain the following regularity result whose obvious proof may be omitted.Corollary 1.4.Let (ρ,u )be a global weak solution to the initial boundary value problem (1.1)–(1.4)with the data satisfying condition (1.7).If there exists a finite positive time T ∗such thatu ∈L s (0,T ∗;L r w )for some (r,s )with2s +3r=1,3<r ≤∞,(1.15)then the solution (ρ,u )satisfies regularity (1.8)for some T >T ∗.A similar result was obtained by Choe and the author [6]assuming,however,astronger condition on u ,that is,u ∈L 4(0,T ∗;D 10).By virtue of Corollary 1.4,we may conclude that the class (which we call a weak Serrin class )in (1.15)is a regularity class for weak solutions of the nonhomogeneous Navier–Stokes equations,which was already proved by Sohr [28]for the homogeneous case.See also a local version of Sohr’s result in [14].Moreover,thanks to a recent result by Dubois [8],the weak Serrin class is a uniqueness class for the homogeneous Navier–Stokes equations.It is also noticed that the same results can be easily derived from regularity and uniqueness results due to Kozono by adapting the arguments in the remarks of Theorem 3in [16].Theorem 1.3and its proof can be used to obtain a global existence result on solutions in the class H 2under some smallness condition on u 0and f (but not on ρ0).Theorem 1.5.For each K >1,there exists a small constant ε>0,depending only on K and the domain Ω,with the following property:if the data ρ0,u 0,and f satisfy|ρ0|L 32∩L∞≤K,|u 0|D 10≤ε,and ∞|f (t )|2L 2dt ≤ε2(1.16)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1421in addition to condition (1.7),then there exists a unique global strong solution (ρ,u )to problem (1.1)–(1.4)satisfying regularity (1.8)for any T <∞.A rather simple proof of Theorem 1.5is provided in section 4.Finally,we recall that in the case when ρ0is bounded away from zero and Ωis a bounded domain in R 3with smooth boundary,Salvi [24]proved the local existence of strong solutions in the class H 2m for every m ≥1.Hence adapting the proofs of Corollary 1.4and Theorem 1.5,we can also prove analogous regularity and global existence results on strong solutions in every class H 2m provided that Ω⊂⊂R 3and ρ0>0on Ω.2.Proof of Theorem 1.3with m =1.In this section,we prove Theorem 1.3for the special case m =1.Let t 0be a fixed time with 0<t 0<T ∗and let us denoteΦ0(T )= T|u (t )|s L r wdt for t 0≤T <T ∗,where (r,s )is any pair satisfying condition (1.14).To prove Theorem 1.3,we haveonly to show that Φ1(T )≤C exp (C exp (C Φ0(T )))for t 0≤T <T ∗.(2.1)Throughout this paper,we denote by C a generic positive constant depending onlyon r ,m ,Φm (t 0),T ∗,Ω,|ρ(0)|L 32∩L ∞,|u (0)|L 6,and the norm of f ,but independent of T in particular.To begin with,we recall from (1.6)that sup0≤t ≤T|ρ(t )|L 32∩L ∞+|√ρu (t )|L 2 +T|u (t )|2D 10dt ≤C(2.2)for t 0≤T <T ∗.2.1.Estimates for T 0|√ρu t (t )|2L 2dt and sup 0≤t ≤T |u (t )|D 10.Next,we willshow thatT 0|√ρu t (t )|2L 2+|u (t )|2D 10∩D 2 dt +sup 0≤t ≤T|u (t )|2D 10≤C exp (C Φ0(T ))(2.3)for t 0≤T <T ∗.To show this,we multiply the momentum equation (1.1)by u t andintegrate over Ω.Then using (1.2)and (1.3),we easily deriveρ|u t |2dx +12d dt |∇u |2dx = ρ(f −u ·∇u )·u t dx andρ|u t |2dx +ddt|∇u |2dx ≤2ρ|f |2dx +2ρ|u ·∇u |2dx.(2.4)On the other hand,since (u,p )is a solution of the stationary Stokes equations−Δu +∇p =Fanddiv u =0inΩ,where F =ρ(f −u ·∇u −u t ),it follows from the classical regularity theory that |∇u |H 1≤C (|F |L 2+|∇u |L 2)(2.5)≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2).1422HYUNSEOK KIMTo estimate the right-hand sides of(2.4)and(2.5),wefirst observe that|u·∇u|L2=|u·∇u|L2,2≤C|u|L rw |∇u|L2rr−2,2,(2.6)which follows from H¨o lder inequality in the Lorenz spaces.See Proposition2.1in[18]. Next,we will show that|∇u|L2rr−2,2≤C|∇u|1−3rL2|∇u|3rH1.(2.7)If r=∞,then(2.7)is obvious.Assuming that3<r<∞,we choose r1and r2suchthat3<r1<r<r2<∞and2r =1r1+1r2.Then in view of H¨o lder and Sobolevinequalities,we have|∇u|L2r ir i−2≤|∇u|1−3r iL2|∇u|3r iL6≤|∇u|1−3r iL2(C|∇u|H1)3r ifor each i=1,2.Since L2r r−2,2is a real interpolation space of L2r2r2−2and L2r1r1−2,moreprecisely,L2r r−2,2=(L2r2r2−2,L2r1r1−2)12,2,it thus follows that|∇u|L2rr−2,2≤C|∇u|12L2r2r2−2|∇u|12L2r1r1−2≤C|∇u|1−3r2L2(C|∇u|H1)3r212|∇u|1−3r1L2(C|∇u|H1)3r112,which proves(2.7).For some facts on the real interpolation theory and Lorenz spaces used above,we refer to sections1.3.3and1.18.6in Triebel’s book[30].The estimates(2.6)and(2.7)yield|u·∇u|L2≤C|u|L rw |∇u|2sL2|∇u|3rH1≤η−3s2r C|u|s2L rw|∇u|L2+η|∇u|H1for any small numberη∈(0,1).Substituting this into(2.5),we obtain|∇u|H1≤C|f|L2+|√ρu t|L2+|u|s2L rw|∇u|L2+|∇u|L2,(2.8)and thus|u·∇u|L2≤η−3s2r C|u|s2L rw+1|∇u|L2+C|f|L2+η|√ρu t|L2.Therefore,substituting this estimate into(2.4)and choosing a sufficiently smallη>0, we conclude that1 2|√ρu t(t)|2L2+ddt|∇u(t)|2L2≤C|f(t)|2L2+C|u(t)|s L rw+1|∇u(t)|2L2(2.9)for t0≤t<T∗.In view of Gronwall’s inequality,we haveT0|√ρu t(t)|2L2dt+sup0≤t≤T|∇u(t)|2L2≤C exp(CΦ0(T))for any T with t0≤T<T∗.Combining this and(2.8),we obtain the desired estimate (2.3).NONHOMOGENEOUS NAVIER–STOKES EQUATIONS14232.2.Estimates for ess sup 0≤t ≤T |√ρu t (t )|2L 2and T 0|u t (t )|2D 10dt .To de-rive these estimates,we differentiate the momentum equation (1.1)with respect to time t and obtainρu tt +ρu ·∇u t −Δu t +∇p t =ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ).Then multiplying this by u t ,integrating over Ω,and using (1.2)and (1.3),we have12d dtρ|u t |2dx + |∇u t |2dx(2.10)=(ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ))·u t dx.Note that since ρ∈C ([0,T ];L 32∩L ∞),ρt ∈C ([0,T ];L 32),and u t ∈L 2(0,T ;D 10)forany T <T ∗,the right-hand side of (2.10)is well defined for almost all t ∈(0,T ∗).Hence using finite differences in time,we can easily show that the identity (2.10)holds for almost all t ∈(0,T ∗).In view of the continuity equation (1.2)again,we deduce from (2.10)that12ddtρ|u t |2dx + |∇u t |2dx≤2ρ|u ||u t ||∇u t |+ρ|u ||u t ||∇u |2+ρ|u |2|u t ||∇2u |(2.11)+ρ|u |2|∇u ||∇u t |+ρ|u t |2|∇u |+ρ|u ||u t ||∇f |+ρ|u ||f ||∇u t |+ρ|f t ||u t |dx ≡8 j =1I j .Following the arguments in [6],we can estimate each term I j :I 1,I 5≤C |ρ|12L ∞|∇u |L 2|√ρu t |L 3|∇u t |L 2≤C |ρ|34L ∞|∇u |L 2|√ρu t |12L 2|∇u t |32L 2≤C |∇u |4L 2|√ρu t |2L 2+116|∇u t |2L 2,I 2,I 3,I 4≤C |ρ|L ∞|∇u |2L 2|∇u t |L 2|∇u |H 1≤C |∇u |4L 2|∇u |2H 1+116|∇u t |2L 2,I 6,I 7≤C |ρ|L 6|∇u |L 2|f |H 1|∇u t |L 2≤C |∇u |2L 2|f |2H 1+116|∇u t |2L 2,and finallyI 8≤C |ρ|L 3|f t |L 2|∇u t |L 2≤C |f t |2L 2+116|∇u t |2L 2.Substitution of these estimates into (2.11)yieldsd dt |√ρu t |2L 2+|∇u t |2L 2≤C |∇u |4L 2 |√ρu t |2L 2+|∇u |2H 1+|f |2H 1 +C |f |2H 1+|f t |2L 2 .1424HYUNSEOK KIMTherefore,by virtue of estimate (2.3),we conclude that ess sup 0≤t ≤T|√ρu t (t )|2L 2+T|∇u t (t )|2L 2dt ≤C exp (C Φ0(T ))(2.12)for t 0≤T <T ∗.On the other hand,using the regularity theory of the Stokesequations again,we have|∇u |H 1≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2)≤C |f |L 2+|√ρu t |L 2+|∇u |32L 2|∇u |12H 1+|∇u |L 2and|∇u |H 1,6≤C (|u t |L 6+|u ·∇u |L 6+|f |L 6+|∇u |L 6)≤C|∇u t |L 2+|∇u |2H1+|f |H 1+|∇u |H 1 .Hence it follows immediately from (2.3)and (2.12)that sup0≤t ≤T|u (t )|2D 10∩D 2+T|∇u (t )|2H 1,6dt ≤C exp (C Φ0(T ))(2.13)for t 0≤T <T ∗.2.3.Estimates for sup 0≤t ≤T |∇ρ(t )|H 1and T0|u (t )|2D 3dt .To derive these,we first observe that each ρx j (j =1,2,3)satisfiesρx jt +u ·∇ρx j =−u x j ·∇ρ.Then multiplying this by ρx j ,integrating over Ω,and summing up,we obtaind dt|∇ρ|2dx ≤C|∇u ||∇ρ|2dx ≤C |∇u |L ∞|∇ρ|2L 2.A similar argument shows thatddt|∇2ρ|2dx ≤C|∇u ||∇2ρ|2+|∇2u ||∇ρ||∇2ρ| dx ≤C |∇u |L ∞|∇2ρ|2L 2+|∇2u |L 6|∇ρ|L 3|∇2ρ|L 2.Hence using Sobolev embedding results and then Gronwall’s inequality,we derive thewell-known estimate|∇ρ(t )|H 1≤C exp C t 0|∇u (τ)|H 1,6dτ≤C exp tC |∇u (τ)|2H 1,6+1 dτ .Therefore by virtue of (2.13),we conclude that sup 0≤t ≤T|∇ρ(t )|H 1≤C exp (C exp (C Φ0(T )))(2.14)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1425for t 0≤T <T ∗.Finally,observing from the regularity theory on the Stokes equations that|u |D 3≤C (|ρu t |H 1+|ρu ·∇u |H 1+|ρf |H 1)≤C (|∇ρ|L 3+1) |∇u t |L 2+|∇u |2H 1+|f |H 1 ,we easily deduce from (2.3),(2.12),(2.13),and (2.14)thatT|u (t )|2D 3dt ≤C exp (C exp (C Φ0(T )))(2.15)for t 0≤T <T ∗.This completes the proof of (2.1)and thus the proof of Theorem 1.3with m =1.3.Proof of Theorem 1.3with m ≥ 2.Assume that m ≥ 2.Then to prove Theorem 1.3,it suffices to show that the following estimate holds for each k ,0≤k <m :Φk +1(T )≤C exp C expC Φk (T )10m for t 0≤T <T ∗.(3.1)The case k =0was already proved in section 2and so it remains to prove (3.1)forthe case 1≤k <m .Let k be a fixed integer with 1≤k <m .From (1.13),we recall thatΦk (T )=1+sup0≤j<ksup0≤t ≤T|∂jt u (t )|D 10∩D 2k −2j +T|∂jt u (t )|2D 2k −2j +1dt(3.2)+sup 0≤t ≤T|∇ρ(t )|H 2k −1+ess sup 0≤t ≤T|√ρ∂kt u (t )|L 2+T|∂kt u (t )|2D 10dtfor any T <T ∗.3.1.Estimates for ∂j t(u ·∇u ),∂j +1t ρ,and ∂jt (ρu )with 0≤j ≤k .To estimate nonlinear terms,we will make repeated use of the following simple lemma whose proof is omitted.Lemma 3.1.If g ∈D 10∩D j ,h ∈H i ,0≤i ≤j ,and j ≥2,thengh ∈H iand|gh |H i ≤C |g |D 10∩D j |h |H i for some constant C >0depending only on j and Ω.Using this lemma together with the fact that∂j t (u·∇u )=j i =0j !i !(j −i )!∂itu ·∇∂j −i t u,we can estimate ∂jt(u ·∇u )as follows:for 0≤j <k ,|∂jt(u ·∇u )|H 2k −2j −1≤Cj i =0|∂it u ·∇∂j −i t u |H 2k −2j −1≤C j i =0|∂it u |D 10∩D 2k −2j |∇∂j −i tu |H 2k −2j −1≤Cj i =0|∂it u |D 10∩D 2k −2i |∂j −i tu |D 10∩D 2k −2(j −i )1426HYUNSEOK KIM and|∂k t(u·∇u)|L2≤C k−1i=0|∂i t u·∇∂k−itu|L2+|∂k t u·∇u|L2≤C k−1i=0|∂i t u|D1∩D2|∇∂k−itu|L2+|∂k t u|D1|∇u|H1≤C k−1i=0|∂i t u|D1∩D2k−2i|∂k−itu|D1+|∂k t u|D1|u|D1∩D2.Hence it follows from(3.2)thatsup 0≤j<ksup0≤t≤T|∂j t(u·∇u)(t)|H2k−2j−1+T|∂k t(u·∇u)(t)|2L2dt≤CΦk(T)4(3.3)for t0≤T<T∗.Applying Lemma3.1to the continuity equationρt=−div(ρu)=−u·∇ρ,(3.4)we also deduce thatsup 0≤t≤T |ρt(t)|H2k−1≤CΦk(T)2for t0≤T<T∗.(3.5)Using(3.4)and(3.5),we can show thatsup 1≤j<ksup0≤t≤T|∂j+1tρ(t)|H2k−2j+T|∂k+1tρ(t)|2L2dt≤CΦk(T)2k+4(3.6)for t0≤T<T∗.A simple inductive proof of(3.6)may be based on the observation that for1≤j<k,|∂j+1tρ|H2k−2j=|−∂j t(u·∇ρ)|H2k−2j≤Cji=0|∂j−itu·∇∂i tρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2j|∇∂itρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2(j−i)|∂itρ|H2k−2(i−1)and|∂k+1t ρ|L2≤Cki=0|∂k−itu·∇∂i tρ|L2≤Cki=0|∂k−itu|D1|∂i tρ|H2.Moreover,it follows easily from(3.6)thatsup 0≤j<ksup0≤t≤T|∂j t(ρu)(t)|H2k−2j+T|∂k t(ρu)(t)|2H1dt≤CΦk(T)4k+10(3.7)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1427for t 0≤T <T ∗.Finally,recalling that∂k +1t f ∈L 2(0,∞;L 2)and ∂jt f ∈L 2(0,∞;H 2k −2j +1)for 0≤j ≤k,we deduce from standard embedding results that∂jt f ∈C ([0,∞);H 2k −2j )for0≤j ≤k.3.2.Estimates for T 0|√ρ∂k +1t u (t )|2L 2dt and sup 0≤t ≤T |∂k t u (t )|D 10.Fromthe momentum equation (1.1),we deriveρ ∂k t u t −Δ∂k t u +∇∂k t p =∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂k t (ρu t ) .Hence multiplying this by ∂k +1t u and integrating over Ω,we haveρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx=∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂kt (ρu t ) ·∂k +1tu dx (3.8)=I 0,1+k j =1k !j !(k −j )!(I j,1+I j,2),whereI j,1=∂j t ρ∂k −j t (f −u ·∇u )·∂k +1t u dx,I j,2=−∂j t ρ∂k −j t u t ·∂k +1t u dx.We easily estimate I 0,1as follows:I 0,1≤|ρ|12L ∞ |∂k t f |L 2+|∂k t (u ·∇u )|L 2 |√ρ∂k +1t u |L 2≤C |∂k t f |2L 2+|∂kt (u ·∇u )|2L2 +12|√ρ∂k +1t u |2L 2.To estimate I j,1for 1≤j ≤k ,we rewrite it asI j,1=d dt∂j t ρ∂k −j t (f −u ·∇u )·∂k t u dx − ∂j +1t ρ∂k −j t (f −u ·∇u )·∂kt u dx −∂j t ρ∂k −j +1t (f −u ·∇u )·∂kt u dx and observe that −∂j +1tρ∂k −j t (f −u ·∇u )·∂kt u dx ≤C |∂k −j t f |2H 1+|∂k −j t (u ·∇u )|2H 1 |∂j +1t ρ|2L 2+|∇∂k t u |2L 2and−∂j tρ∂k −j +1t (f −u ·∇u )·∂k t u dx ≤C |∂j t ρ|2H 1 |∂k −j +1t f |2L 2+|∂k −j +1t (u ·∇u )|2L 2 +|∇∂k t u |2L 2.1428HYUNSEOK KIMUsing the continuity equation (1.2),we can also estimate I j,2as follows:I 1,2=− ρt 12|∂k t u |2 t dx =−d dt ρt 12|∂k t u |2dx + ∂2t ρ12|∂k t u |2dx =−d dt ρu ·∇ 12|∂k t u |2 dx + ∂t (ρu )·∇ 12|∂k tu |2dx ≤−d dt (ρu ·∇∂k t u )·∂k t u dx +C |∂t (ρu )|H 1|∇∂k t u |2L 2and similarlyI j,2=−d dt ∂j t ρ∂k −j t u t ·∂k t u dx + ∂j +1t ρ∂k −j t u t +∂j t ρ∂k −j +1t u t ·∂kt u dx ≤−d dt ∂j −1t (ρu )·∇ ∂k −j +1t u ·∂kt u dx +C |∂j t (ρu )|H 1|∇∂k −j +1t u |L 2+|∂j −1t (ρu )|H 1|∇∂k −j +2t u |L 2 |∇∂kt u |L 2for 2≤j ≤k .Substituting all the estimates into (3.8),we have12 ρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx ≤d dt ⎛⎝k j =1k !j !(k −j )!∂j t ρ∂k −j t (f −u ·∇u )·∂k t u −(ρu ·∇∂k t u )·∂k t u ⎞⎠dx −d dt kj =2k !j !(k −j )!∂j −1t(ρu )·∇ ∂k −j +1t u ·∂kt u dx+Ck −1 j =1|∂jt (ρu )|2H 1+|∂j +1t ρ|2H 1|∂k −j t f |2H 1+|∂k −jt (u·∇u )|2H 1+|∂k −j +1t u |2D 1+C |∂k +1t ρ|2L 2|f |2H 1+|u |2D 10∩D 2+C 1+|∂t ρ|2H 1 |∂k t f |2L 2+|∂k t (u ·∇u )|2L2 +C |∂k t (ρu )|2H 1+C 1+|∂t (ρu )|2H 1+|∇∂t u |2L 2 |∇∂k t u |2L 2.Hence,integrating this in time over (t 0,T )and using (3.3),(3.5),(3.6),and (3.7)together with the estimates|∂j t ρ||∂k −j t (f −u ·∇u )||∂k t u |dx≤η−1|∂j t ρ|2H 1|∂k −j t (f −u ·∇u )|2L 2+η|∇∂k t u |L 2,ρ|u ||∇∂k t u ||∂k t u |dx ≤η−3C |ρ|3L ∞|∇u |4L 2|√ρ∂k t u |2L 2+η|∇∂k t u |2L 2and|∂j −1t (ρu )||∇∂k −j +1t u ||∂kt u |+|∂k −j +1t u ||∇∂kt u |dx≤η−1C |∂j −1t (ρu )|2H 1|∇∂k −j +1tu |2L 2+η|∇∂kt u |L 2,NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1429 whereηis any small positive number,we deduce thatTt0|√ρ∂k+1tu(t)|2L2dt+|∇∂k t u(T)|2L2≤CΦk(T)20m+CTt01+|∂t(ρu)(t)|2H1+|u t(t)|2D1|∇∂k t u(t)|2L2dtfor t0≤T<T∗.Note thatT t01+|∂t(ρu)(t)|2H1+|u t(t)|2D1dt≤CΦk(T)10m.Therefore,in view of Gronwall’s inequality,we conclude that T0|√ρ∂k+1tu(t)|2L2dt+sup0≤t≤T|∂k t u(t)|2D1≤C expCΦk(T)10m(3.9)for any T with t0≤T<T∗.3.3.Estimates for ess sup0≤t≤T|√ρ∂k+1tu(t)|L2andT|∂k+1tu(t)|2D1dt.From the momentum equation(1.1),it follows thatρ∂k+1tut+ρu·∇∂k+1tu−Δ∂k+1tu+∇∂k+1tp=∂k+1t(ρf)+ρ∂k+1tu t−∂k+1t(ρu t)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u).Multiplying this by∂k+1tu and integrating overΩ,we have1 2ddtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx=∂k+1t(ρf)·∂k+1tu dx+ρ∂k+1tu t−∂k+1t(ρu t)·∂k+1tu dx(3.10)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u)·∂k+1tu dx.This identity can be derived rigorously by using a standardfinite difference method because if0<T<T∗,then∂m tρ∈L2(0,T;L32∩L2)and∂j tρ∈C([0,T];L32∩L∞) for0≤j<m.Thefirst term of the right-hand side in(3.10)is bounded byCkj=0|∂j tρ||∂k−j+1tf||∂k+1tu|dx+|∂k+1tρ||f||∂k+1tu|dx≤Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1+16|∇∂k+1tu|2L2.In view of the continuity equation(1.2),we can rewrite the second term as−k+1j=1(k+1)!j!(k−j+1)!∂j tρ∂k−j+1tu t·∂k+1tu dx=−k+1j=1(k+1)!j!(k−j+1)!∂j−1t(ρu)·∇∂k−j+2tu·∂k+1tudx,1430HYUNSEOK KIM which is bounded byC|ρ|L∞|u|2D10∩D2|√ρ∂k+1tu|2L2+Ckj=1|∂j t(ρu)|2H1|∂k−j+1tu|2D1+16|∇∂k+1tu|2L2.Finally,the last term is bounded byCkj=1|∂j t(ρu)||∇∂k−j+1tu||∂k+1tu|dx+|∂k+1t(ρu)||∇u||∂k+1tu|dx≤Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2+C|ρ|L∞|u|2D1∩D2|√ρ∂k+1tu|2L2+1|∇∂k+1tu|2L2.Hence substituting these estimates into(3.10),we haved dtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx≤C1+|u|2D1∩D2|√ρ∂k+1tu|2L2+Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1 +Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2.Therefore,by virtue of(3.5),(3.6),(3.7),and(3.9),we conclude thatess sup0≤t≤T |√ρ∂k+1tu(t)|L2+T|∂k+1tu(t)|2D1dt≤C expCΦk(T)10m(3.11)for any T with t0≤T<T∗.3.4.Estimates for sup0≤t≤T|∂j t u(t)|D10∩D2k−2j+2with0≤j≤k.Toderive these estimates,we observe that∂j t u∈C([0,T∗);D10,σ)and−Δ∂j t u+∇∂j t p=∂j t(ρf−ρu·∇u−ρu t) (3.12)for each j≤k.From(3.5),(3.6),(3.9),and(3.11),it follows easily thatess sup0≤t≤T|∂k t(ρu·∇u)(t)|L2+|∂k t(ρu t)(t)|L2≤C expCΦk(T)10mfor t0≤T<T∗.Hence applying the regularity theory of the Stokes equations to (3.12)with j=k,we obtainsup 0≤t≤T |∂k t u(t)|D1∩D2≤C expCΦk(T)10mfor t0≤T<T∗.It also follows from the Stokes regularity theory that for0≤j<k,|∂j t u|D10∩D2k−2j+2≤C|∂jt(ρf−ρu·∇u−ρu t)|H2k−2j+C|∂j t u|D1.(3.13)。