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
The main well known observation is
“all the linear terms are divergence free,
the inertial nonlinear term not”
PLv = v
PLtv = tv
PLp = 0
tv+ Sv+PL((v.)v)= PLf
 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 very promising g.i.c., further perspectives
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