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

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An alternative approach in the theory of 3D
incompressible Navier-Stokes equations
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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.
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Navier-Stokes
Joint research with Jiri NEUSTUPA
(Prague) References
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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
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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

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Navier-Stokes
PLv v PL?tv ?tv PL?p 0 ?tv? SvPL((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 (?)
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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

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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)

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Navier-Stokes
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Navier-Stokes
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Navier-Stokes
(1) - (4)
? existence of weak solutions as for the
standard model remark
...
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Navier-Stokes
(1) - (4)
? Similar theorem for the standard model
belongs to the classical theory of NSeqs
remark with j 2
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(1) - (4)
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(1) - (4)
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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
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Navier-Stokes
? We can finally arrive at the inequality
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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
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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