The Navier-Stokes Problem in the 21st Century
The Navier-Stokes Problem in the 21st Century
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DOI：
10.1201/b19556
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发表时间：
2016-03
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通讯作者：
P. Lemarié–Rieusset
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作者：
P. Lemarié–Rieusset
粘土千禧奖颁奖仪式三维流体流动的规律性：21世纪的数学挑战克莱千年奖克莱千年奖的Navier-Stokes方程边界和Navier-Stokes克莱千年问题的物理意义Stokes方程参考系对流定理质量守恒牛顿第二定律压力应变应力流体动力学Navier-Stokes方程涡度边界项爆炸湍流方程的历史科学革命时代的力学Bernoulli's Hydrodymica D 'Alembert Euler Laplacian物理学Navier，Cauchy，Poisson，Saint-Venant，and Stokes Reynolds Oseen，Leray，Hopf，和Ladyzhenskaya湍流模型经典解热核Poisson方程Helmholtz分解Stokes方程Oseen张量Navier-Stokes问题的经典解小数据和整体解整体解的时间渐近性稳态解空间渐近性涡量的空间渐近性中间结论Navier-Stokes积分方程的容性方法积分Navier-Banach空间中的Stokes问题二次积分方程齐型控制函数空间上的广义Riesz势Navier-Stokes积分方程Osen定理的控制函数证明泛函空间和乘子微分和积分Navier-Stokes方程一致局部估计热方程Stokes方程Osen方程Navier-Stokes方程的非常弱解Navier-Stokes方程的温和解Navier-Stokes方程的合适解Stokes方程Lebesgue或Sobolev空间中的温和解Kato的温和解Hilbert空间中的局部解Hilbert空间中的整体解Sobolev空间中的交换子估计Lebesgue空间极大函数关于真实的插值空间的基本引理L3的唯一性解Besov或Morrey空间中的Mild解Morrey空间Morrey空间与极大函数Morrey解的唯一性Besov空间正则Besov空间Triebel-Lizorkin空间Fourier变换与Navier-Stokes方程空间BMO-1与Koch和Tataru定理Koch和Tataru定理Q-空间BMO-1不适定性的一个特殊子类适定性温和解的大数据整体解的稳定性解析性小数据解的特例Navier-Stokes方程的对称性二维半流动轴对称解螺旋解Brandolese对称解自相似解Navier-Stokes方程的定常解朗道解时间周期解贝尔特拉米流爆炸？第一个准则廉价的Navier-Stokes方程的爆破Serrin准则Serrin准则的一些进一步推广涡度平方Leray弱解Rellich引理Leray弱解弱-强唯一性：Prodi-Serrin准则乘积空间R x R3上的强唯一性和Morrey空间几乎强解弱扰动弱解的部分正则性结果弱解的内部正则性Serrin's关于内部正则性的定理关于内部正则性的O 'Leary定理关于抛物Morrey空间的进一步结果Hausdorff测度奇异时间局部能量不等式关于部分正则性的Caffarelli-Kohn-Nirenberg定理Caffarelli-Kohn-Nirenberg准则的证明奇异点集的抛物Hausdorff维数关于压力在Caffarelli，Kohn，和Nirenberg正则性定理一致局部L2解一致局部平方可积解局部Leray解的局部不等式Caffarelli，Kohn，和Nirenberg ε-正则性准则强唯一性结果适当解的L3理论L3中初值的局部Leray解L3中Cauchy问题爆破的临界元局部Leray解的向后唯一性Seregin定理有外力存在下Navier-Stokes方程Cauchy问题的已知结果适当解的局部估计适当解的唯一性Caffarelli-Kohn-Nirenberg正则性准则的定量单尺度估计适当解集的拓扑结构Escauriaza，Seregin，和Sverak定理自相似性和Leray-Schauder原理Leray-Schauder原理稳态解贾和Sverak定理的自相似性声明局部有界初始数据的情况粗糙数据的情况非后向自相似解的存在性α-模型全局存在性，近似方程的唯一性和收敛性问题Leray's软化和Leray-alpha模型Navier-Stokes alpha模型Clark- alpha模型简化的Bardina模型Reynolds张量Navier-Stokes方程的其他近似Faedo-Galerkin近似频率截止高粘度Ladyzhenskaya模型阻尼Navier-Stokes方程人工可压缩性Temam模型Vishik和Fursikov模型双曲近似结论能量不等式温和解的临界空间（势）爆破模型临界元法标记和词汇参考书目索引
Presentation of the Clay Millennium Prizes Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century The Clay Millennium Prizes The Clay Millennium Prize for the Navier-Stokes equations Boundaries and the Navier-Stokes Clay Millennium Problem The physical meaning of the Navier-Stokes equations Frames of references The convection theorem Conservation of mass Newton's second law Pressure Strain Stress The equations of hydrodynamics The Navier-Stokes equations Vorticity Boundary terms Blow up Turbulence History of the equation Mechanics in the Scientific Revolution era Bernoulli's Hydrodymica D'Alembert Euler Laplacian physics Navier, Cauchy, Poisson, Saint-Venant, and Stokes Reynolds Oseen, Leray, Hopf, and Ladyzhenskaya Turbulence models Classical solutions The heat kernel The Poisson equation The Helmholtz decomposition The Stokes equation The Oseen tensor Classical solutions for the Navier-Stokes problem Small data and global solutions Time asymptotics for global solutions Steady solutions Spatial asymptotics Spatial asymptotics for the vorticity Intermediate conclusion A capacitary approach of the Navier-Stokes integral equations The integral Navier-Stokes problem Quadratic equations in Banach spaces A capacitary approach of quadratic integral equations Generalized Riesz potentials on spaces of homogeneous type Dominating functions for the Navier-Stokes integral equations A proof of Oseen's theorem through dominating functions Functional spaces and multipliers The differential and the integral Navier-Stokes equations Uniform local estimates Heat equation Stokes equations Oseen equations Very weak solutions for the Navier-Stokes equations Mild solutions for the Navier-Stokes equations Suitable solutions for the Navier-Stokes equations Mild solutions in Lebesgue or Sobolev spaces Kato's mild solutions Local solutions in the Hilbertian setting Global solutions in the Hilbertian setting Sobolev spaces A commutator estimate Lebesgue spaces Maximal functions Basic lemmas on real interpolation spaces Uniqueness of L3 solutions Mild solutions in Besov or Morrey spaces Morrey spaces Morrey spaces and maximal functions Uniqueness of Morrey solutions Besov spaces Regular Besov spaces Triebel-Lizorkin spaces Fourier transform and Navier-Stokes equations The space BMO-1 and the Koch and Tataru theorem Koch and Tataru's theorem Q-spaces A special subclass of BMO-1 Ill-posedness Further results on ill-posedness Large data for mild solutions Stability of global solutions Analyticity Small data Special examples of solutions Symmetries for the Navier-Stokes equations Two-and-a-half dimensional flows Axisymmetrical solutions Helical solutions Brandolese's symmetrical solutions Self-similar solutions Stationary solutions Landau's solutions of the Navier-Stokes equations Time-periodic solutions Beltrami flows Blow up? First criteria Blow up for the cheap Navier-Stokes equation Serrin's criterion Some further generalizations of Serrin's criterion Vorticity Squirts Leray's weak solutions The Rellich lemma Leray's weak solutions Weak-strong uniqueness: the Prodi-Serrin criterion Weak-strong uniqueness and Morrey spaces on the product space R x R3 Almost strong solutions Weak perturbations of mild solutions Partial regularity results for weak solutions Interior regularity Serrin's theorem on interior regularity O'Leary's theorem on interior regularity Further results on parabolic Morrey spaces Hausdorff measures Singular times The local energy inequality The Caffarelli-Kohn-Nirenberg theorem on partial regularity Proof of the Caffarelli-Kohn-Nirenberg criterion Parabolic Hausdorff dimension of the set of singular points On the role of the pressure in the Caffarelli, Kohn, and Nirenberg regularity theorem A theory of uniformly locally L2 solutions Uniformly locally square integrable solutions Local inequalities for local Leray solutions The Caffarelli, Kohn, and Nirenberg epsilon-regularity criterion A weak-strong uniqueness result The L3 theory of suitable solutions Local Leray solutions with an initial value in L3 Critical elements for the blow up of the Cauchy problem in L3 Backward uniqueness for local Leray solutions Seregin's theorem Known results on the Cauchy problem for the Navier-Stokes equations in presence of a force Local estimates for suitable solutions Uniqueness for suitable solutions A quantitative one-scale estimate for the Caffarelli-Kohn-Nirenberg regularity criterion The topological structure of the set of suitable solutions Escauriaza, Seregin, and Sverak's theorem Self-similarity and the Leray-Schauder principle The Leray-Schauder principle Steady-state solutions Self-similarity Statement of Jia and Sverak's theorem The case of locally bounded initial data The case of rough data Non-existence of backward self-similar solutions alpha-models Global existence, uniqueness and convergence issues for approximated equations Leray's mollification and the Leray-alpha model The Navier-Stokes alpha -model The Clark- alpha model The simplified Bardina model Reynolds tensor Other approximations of the Navier-Stokes equations Faedo-Galerkin approximations Frequency cut-off Hyperviscosity Ladyzhenskaya's model Damped Navier-Stokes equations Artificial compressibility Temam's model Vishik and Fursikov's model Hyperbolic approximation Conclusion Energy inequalities Critical spaces for mild solutions Models for the (potential) blow up The method of critical elements Notations and glossary Bibliography Index