51 research outputs found
A KAM approach to the inviscid limit for the 2D Navier-Stokes equations
In this paper we investigate the inviscid limit for
time-quasi-periodic solutions of the incompressible Navier-Stokes equations on
the two-dimensional torus , with a small time-quasi-periodic
external force. More precisely, we construct solutions of the forced Navier
Stokes equation, bifurcating from a given time quasi-periodic solution of the
incompressible Euler equations and admitting vanishing viscosity limit to the
latter, uniformly for all times and independently of the size of the external
perturbation. Our proof is based on the construction of an approximate
solution, up to an error of order and on a fixed point argument
starting with this new approximate solution. A fundamental step is to prove the
invertibility of the linearized Navier Stokes operator at a quasi-periodic
solution of the Euler equation, with smallness conditions and estimates which
are uniform with respect to the viscosity parameter. To the best of our
knowledge, this is the first positive result for the inviscid limit problem
that is global and uniform in time and it is the first KAM result in the
framework of the singular limit problems
Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems
International audienceWe provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths. Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption. Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form
Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves
This paper is concerned with a priori regularity for
three-dimensional doubly periodic travelling gravity waves whose fundamental
domain is a symmetric diamond. The existence of such waves was a long standing
open problem solved recently by Iooss and Plotnikov. The main difficulty is
that, unlike conventional free boundary problems, the reduced boundary system
is not elliptic for three-dimensional pure gravity waves, which leads to small
divisors problems. Our main result asserts that sufficiently smooth diamond
waves which satisfy a diophantine condition are automatically . In
particular, we prove that the solutions defined by Iooss and Plotnikov are
. Two notable technical aspects are that (i) no smallness condition
is required and (ii) we obtain an exact paralinearization formula for the
Dirichlet to Neumann operator.Comment: Corrected versio
Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems
The paper concerns boundary value problems for general nonautonomous first
order quasilinear hyperbolic systems in a strip. We construct small global
classical solutions, assuming that the right hand side is small. For the
nonhomogeneous version of a linearized problem, we provide stable dissipativity
conditions ensuring a unique bounded continuous solution for any smooth
right-hand side. In the autonomous case, this solution is two times
continuously differentiable. In the nonautonomous case, the continuous solution
is differentiable under additional dissipativity conditions, which are
essential. A crucial ingredient of our approach is a perturbation theorem for
general linear hyperbolic systems. In the case that all data of the quasilinear
problem are almost periodic, we prove that the bounded solution is also almost
periodic.Comment: 42 pages. The main result is generalized to cover more general first
order quasilinear hyperbolic systems; improved presentation, a correction in
Example in Subsection 3.
Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation
We study the nonlinear inhomogeneous wave equation in one space dimension:
. By constructing some "decoupled" Riccati type
equations for smooth solutions, we provide a singularity formation result
without restrictions on the total variation of unknown, which generalize
earlier singularity results of Lax and the first author. These results are
applied to several one-dimensional hyperbolic models, such as compressible
Euler flows with a general pressure law, elasticity in an inhomogeneous medium,
transverse MHD flow, and compressible flow in a variable area duct
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