Navier–Stokes equations

About: Navier–Stokes equations is a research topic. Over the lifetime, 18180 publications have been published within this topic receiving 552555 citations. The topic is also known as: Navier-Stokes equations.

The Navier-Stokes Problem in the 21st Century

Abstract: 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

On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities

Abstract: We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier–Stokes/Cahn–Hilliard system, which is capable of describing the evolution of droplet formation and collision during the flow. We prove the existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures—in contrast to the usual case of a smooth free energy density—that the concentration stays in the physical reasonable interval. Furthermore, we find that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn–Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a Stokes system with variable viscosity and forces in L2(0, ∞; Hs(Ω)), \({s \in [0, \frac12)}\) , which are new themselves.

Entropy Stable Spectral Collocation Schemes for the Navier--Stokes Equations: Discontinuous Interfaces

Abstract: Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier--Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier--Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy.

Some Mathematical Problems in Geophysical Fluid Dynamics

Abstract: This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions 2 and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 1990s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction ( Section 1.4 ).