111 research outputs found
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
The goal of this note is to show that, also in a bounded domain , with , any weak solution,
, of the Euler equations of ideal incompressible fluid in
, with the
impermeability boundary condition: on
, is of constant energy on the interval
provided the velocity field , with $\alpha>\frac13\,.
The Incompressible Euler Limit of the Boltzmann Equation with Accommodation Boundary Condition
The convergence of solutions of the incompressible Navier-Stokes equations
set in a domain with boundary to solutions of the Euler equations in the large
Reynolds number limit is a challenging open problem both in 2 and 3 space
dimensions. In particular it is distinct from the question of existence in the
large of a smooth solution of the initial-boundary value problem for the Euler
equations. The present paper proposes three results in that direction. First,
if the solutions of the Navier-Stokes equations satisfy a slip boundary
condition with vanishing slip coefficient in the large Reynolds number limit,
we show by an energy method that they converge to the classical solution of the
Euler equations on its time interval of existence. Next we show that the
incompressible Navier-Stokes limit of the Boltzmann equation with Maxwell's
accommodation condition at the boundary is governed by the Navier-Stokes
equations with slip boundary condition, and we express the slip coefficient at
the fluid level in terms of the accommodation parameter at the kinetic level.
This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat.
Phys. 63 (1991), 323-344]. Finally, we establish the incompressible Euler limit
of the Boltzmann equation set in a domain with boundary with Maxwell's
accommodation condition assuming that the accommodation parameter is small
enough in terms of the Knudsen number. Our proof uses the relative entropy
method following closely the analysis in [L. Saint-Raymond, Arch. Ration. Mech.
Anal. 166 (2003), 47-80] in the case of the 3-torus, except for the boundary
terms, which require special treatment.Comment: 40 page
Short time heat diffusion in compact domains with discontinuous transmission boundary conditions
We consider a heat problem with discontinuous diffusion coefficientsand
discontinuous transmission boundary conditions with a resistancecoefficient.
For all compact -domains with a
-set boundary (for instance, aself-similar fractal), we find the first term
of the small-timeasymptotic expansion of the heat content in the complement
of, and also the second-order term in the case of a regularboundary.
The asymptotic expansion is different for the cases offinite and infinite
resistance of the boundary. The derived formulasrelate the heat content to the
volume of the interior Minkowskisausage and present a mathematical
justification to the de Gennes'approach. The accuracy of the analytical results
is illustrated bysolving the heat problem on prefractal domains by a finite
elementsmethod
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