An alternative approach in the theory of 3D incompressible Navier-Stokes equations

1. An alternative approach in the theory of 3D incompressible Navier-Stokes equations

2. Navier-Stokes The questions addressed in this lecture are motivated by some recent results concerning the regularity of weak solutions to the 3D-Navier-Stokes equations to justify a global in time regular solution represents a challenging problem.

3. Navier-Stokes Joint research with Jiri NEUSTUPA (Prague) References [1] [2] [3] [4]

4. Navier-Stokes 3D viscous incompressible fluids The “particle-fluid” through a point x at time t is characterized by a constant mass density, r = 1, and a vector field v = v(t,x), describing the velocity, solution to the nonlinear system the motion, driven by f(.,.), the external forces

5. Navier-Stokes • p = p(x,t), a scalar field, is the associated pressure • in fact determined by the velocity v • (uniquely up to an additive constant) • no boundary condition for p • no initial condition for p • p diagnostic variable [see application in meteorology, compare with v pronostic variable]

6. Navier-Stokes PLv = v PLtv = tv PLp = 0 tv+ Sv+PL((v.)v)= PLf The main well known observation is “all the linear terms are divergence free, the inertial nonlinear term not”  the Stokes operator S = -PLis not the Laplace operator  the main challenge is “the question of existence and uniqueness of solutions v to NSeqs is still open”  the related questions concern possible irregular solutions (?)

7. Navier-Stokes • Discussion about the boundary conditions • no slip relevant to flows • (homogeneous Dirichlet type) in bounded domains • [impermeable walls] • space periodic relevant to idealized flows • (~absence of bdry conditions) far away from physical real boundaries • g.i.c. relevant to describe flows with • “generalized impermeability conditions” tangential behavior of v, curl v, ... • (vorticity type) at the boundary

8. Navier-Stokes • Discussion about the boundary conditions • no slip  a « standard » model • (homogeneous Dirichlet type) with • space periodic • (~absence of bdry conditions) • g.i.c.  a « natural » model • “generalized impermeability conditions” with • (vorticity type)

9. Navier-Stokes

10. Navier-Stokes

11. Navier-Stokes

12. Navier-Stokes

13. Navier-Stokes (1) - (4) existence of weak solutions as for the “standard” model + remark ...

14. Navier-Stokes (1) - (4)  Similar theorem for the “standard” model belongs to the classical theory of NSeqs + remark with j = 2

15. Navier-Stokes (1) - (4)

16. Navier-Stokes (1) - (4)

17. Navier-Stokes Stokes operator in our case, S = A 2  Last integral equals zero (int. by parts, continuity eq.) Integral on the boundary represents the crucial point Second integral can be treated with a diagonal representation of tensor s

18. Navier-Stokes  We can finally arrive at the inequality

19. Navier-Stokes Geometrical choice, and regularity up to the boundary Then the surface integrals equals zero, and the inequality for the global enstrophy becomes an equality Then the surface integral is estimates as previously, and the enstrophy inequality holds in the form

20. Navier-Stokes In our previous papers, we have shown that the interior regularity of a so-called suitable weak solution v to the Navier-Stokes equations can be guaranteed by similar integrability conditions on one of the eigenvalues of the symmetrized gradient of v. The main reason why these results were proved only locally, in the interior of the domain , was 1. v assumed to satisfy the homogeneous Dirichlet boundary condition, in which case 2. we were not able to perform necessary integrations by parts on the whole domain, then the idea to localize the « standard » model around a possible singular point using strongly the information on the 1-dimensional Haussdorff measure of the set of possible singular points provided by Caffarelli-Kohn-Nirenberg … Here we have extended the results of regularity up to the boundary for the « natural » model (assuming for v the generalized impermeability conditions at the boundary), - first using a simple structure of the boundary of a cube, - secondly estimating the surface integrals for a bounded simply connected domain very promising g.i.c., further perspectives