2,105 research outputs found
Soft congestion approximation to the one-dimensional constrained Euler equations
This article is concerned with the analysis of the one-dimensional
compressible Euler equations with a singular pressure law, the so-called hard
sphere equation of state. The result is twofold. First, we establish the
existence of bounded weak solutions by means of a viscous regularization and
refined compensated compactness arguments. Second, we investigate the smooth
setting by providing a detailed description of the impact of the singular
pressure on the breakdown of the solutions. In this smooth framework, we
rigorously justify the singular limit towards the free-congested Euler
equations, where the compressible (free) dynamics is coupled with the
incompressible one in the constrained (i.e. congested) domain
Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation
This article is concerned with the asymptotic behavior of the two-dimensional
inviscid Boussinesq equations with a damping term in the velocity equation.
Precisely, we provide the time-decay rates of the smooth solutions to that
system. The key ingredient is a careful analysis of the Green kernel of the
linearized problem in Fourier space, combined with bilinear estimates and
interpolation inequalities for handling the nonlinearity
Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition
In the context of hyperbolic systems of balance laws, the Shizuta-Kawashima
coupling condition guarantees that all the variables of the system are
dissipative even though the system is not totally dissipative. Hence it plays a
crucial role in terms of sufficient conditions for the global in time existence
of classical solutions. However, it is easy to find physically based models
that do not satisfy this condition, especially in several space dimensions. In
this paper, we consider two simple examples of partially dissipative hyperbolic
systems violating the Shizuta-Kawashima condition ([SK]) in 3D, such that some
eigendirections do not exhibit dissipation at all. We prove that, if the source
term is non resonant (in a suitable sense) in the direction where dissipation
does not play any role, then the formation of singularities is prevented,
despite the lack of dissipation, and the smooth solutions exist globally in
time. The main idea of the proof is to couple Green function estimates for
weakly dissipative hyperbolic systems with the space-time resonance analysis
for dispersive equations introduced by Germain, Masmoudi and Shatah. More
precisely, the partially dissipative hyperbolic systems violating [SK] are
endowed, in the non-dissipative directions, with a special structure of the
nonlinearity, the so-called Nonresonant Bilinear Form for the wave equation
(see Pusateri and Shatah, CPAM 2013)
A two-dimensional flea on the elephant phenomenon and its numerical visualization
First Published in Multiscale Modeling and Simulation in 17.1 (2019): 137-166, published by the Society for Industrial and Applied Mathematics (SIAM)Localization phenomena (sometimes called flea on the elephant) for the operator Lvarepsilon = varepsilon 2Δ u + p(x)u, p(x) being an asymmetric double well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for Lvarepsilon, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layersEnrique Zuazua’s research was supported by the Advanced
Grant DyCon (Dynamical Control) of the European Research Council Executive
Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO
(Spain) and the ICON project of the French ANR-16-ACHN-0014. L.G. thanks Profs.
François Bouchut and Roberto Natalini for some technical discussion
Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows
Significant advancements have emerged in the theory of asymptotic stability
of shear flows in stably stratified fluids. In this comprehensive review, we
spotlight these recent developments, with particular emphasis on novel
approaches that exhibit robustness and applicability across various contexts.Comment: This is a note/review for the seminar Laurent Schwart
Strong ill-posedness in of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations
We prove the strong ill-posedness in the sense of Hadamard of the
two-dimensional Boussinesq equations in without
boundary, extending to the case of systems the method that Shikh Khalil \&
Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large
class of initial data with velocity and density of small size, for which the horizontal density gradient has a
strong -norm inflation in infinitesimal time, while
the vorticity and the vertical density gradient remain bounded. Furthermore,
exploiting the three dimensional version of Elgindi's decomposition of the
Biot-Savart law, we apply the method to the three-dimensional axisymmetric
Euler equations with swirl and away from the vertical axis, showing that a
large class of initial data with velocity field uniformly bounded in provides a solution whose swirl component has a strong
-norm inflation in infinitesimal time, while the
potential vorticity remains bounded at least for small times. Finally, the
-norm inflation of the swirl (and the vorticity field) is quantified
from below by an explicit lower bound which depends on time, the size of the
data and it is valid for small times
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