3,716 research outputs found
Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics
We investigate non-contraction of large perturbations around intermediate
entropic shock waves and contact discontinuities for the three-dimensional
planar compressible isentropic magnetohydrodynamics (MHD). To do that, we take
advantage of criteria developed by Kang and Vasseur in [6], and non-contraction
property is measured by pseudo distance based on relative entropy
Contraction property for large perturbations of shocks of the barotropic Navier-Stokes system
This paper is dedicated to the construction of a pseudo-norm, for which small
shock profiles of the barotropic Navier-Stokes equation have a contraction
property. This contraction property holds in the class of any large 1D weak
solutions to the barotropic Navier-Stokes equation. It implies a stability
condition which is independent of the strength of the viscosity. The proof is
based on the relative entropy method, and is reminiscent to the notion of
a-contraction first introduced by the authors in the hyperbolic case.Comment: Accepted in J. Eur. Math. Soc. (JEMS
A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment
We consider the kinetic Cucker-Smale model with local alignment as a
mesoscopic description for the flocking dynamics. The local alignment was first
proposed by Karper, Mellet and Trivisa \cite{K-M-T-3}, as a singular limit of a
normalized non-symmetric alignment introduced by Motsch and Tadmor
\cite{M-T-1}. The existence of weak solutions to this model is obtained in
\cite{K-M-T-3}. The time-asymptotic flocking behavior is shown in this article.
Our main contribution is to provide a rigorous derivation from mesoscopic to
mascroscopic description for the Cucker-Smale flocking models. More precisely,
we prove the hydrodynamic limit of the kinetic Cucker-Smale model with local
alignment towards the pressureless Euler system with nonlocal alignment, under
a regime of strong local alignment. Based on the relative entropy method, a
main difficulty in our analysis comes from the fact that the entropy of the
limit system has no strictly convexity in terms of density variable. To
overcome this, we combine relative entropy quantities with the 2-Wasserstein
distance
Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions
We study the global existence and uniqueness of weak solutions to kinetic
Kolmogorov-Vicsek models that can be considered a non-local non-linear
Fokker-Planck type equation describing the dynamics of individuals with
orientational interactions. This model is derived from the discrete
Couzin-Vicsek algorithm as mean-field limit \cite{B-C-C,D-M}, which governs the
interactions of stochastic agents moving with a velocity of constant magnitude,
i.e. the the corresponding velocity space for these type of Kolmogorov-Vicsek
models are the unit sphere. Our analysis for estimates and compactness
properties take advantage of the orientational interaction property meaning
that the velocity space is a compact manifold
-contraction for shock waves of scalar viscous conservation laws
We consider the -contraction up to a shift for viscous shocks of scalar
viscous conservation laws with strictly convex fluxes in one space dimension.
In the case of a flux which is a small perturbation of the quadratic burgers
flux, we show that any viscous shock induces a contraction in , up to a
shift. That is, the norm of the difference of any solution of the viscous
conservation law, with an appropriate shift of the shock wave, does not
increase in time. If, in addition, the difference between the initial value of
the solution and the shock wave is also bounded in , the norm of the
difference converges at the optimal rate . Both results do not
involve any smallness condition on the initial value, nor on the size of the
shock. In this context of small perturbations of the quadratic Burgers flux,
the result improves the Choi and Vasseur's result in [7]. However, we show that
the -contraction up to a shift does not hold for every convex flux. We
construct a smooth strictly convex flux, for which the -contraction does
not hold any more even along any Lipschitz shift
Criteria on contractions for entropic discontinuities of systems of conservation laws
We study the contraction properties (up to shift) for admissible
Rankine-Hugoniot discontinuities of systems of conservation laws
endowed with a convex entropy. We first generalize the criterion developed in
[47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our
generalized criterion guarantees the contraction property for extremal shocks
of a large class of systems, including the Euler system. Moreover, we introduce
necessary conditions for contraction, specifically targeted for intermediate
shocks. As an application, we show that intermediate shocks of the
two-dimensional isentropic magnetohydrodynamics do not verify any of our
contraction properties. We also investigate the contraction properties, for
contact discontinuities of the Euler system, for a certain range of contraction
weights.
All results do not involve any smallness condition on the initial
perturbation, nor on the size of the shock
Asymptotic analysis of Vlasov-type equations under strong local alignment regime
We consider the hydrodynamic limit of a collisionless and non-diffusive
kinetic equation under strong local alignment regime. The local alignment is
first considered by Karper, Mellet and Trivisa in [24], as a singular limit of
an alignment force proposed by Motsch and Tadmor in [32]. As the local
alignment strongly dominate, a weak solution to the kinetic equation under
consideration converges to the local equilibrium, which has the form of
mono-kinetic distribution. We use the relative entropy method and weak
compactness to rigorously justify the weak convergence of our kinetic equation
to the pressureless Euler system
Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow
We consider the so-called spatially homogenous Kolmogorov-Vicsek model, a
non-linear Fokker-Planck equation of self-driven stochastic particles with
orientation interaction under the space-homogeneity. We prove the global
existence and uniqueness of weak solutions to the equation. We also show that
weak solutions exponentially converge to a steady state, which has the form of
the Fisher-von Mises distribution
-type contraction for shocks of scalar viscous conservation laws with strictly convex flux
We study the -type contraction property of large perturbations around
shock waves of scalar viscous conservation laws with strictly convex fluxes in
one space dimension. The contraction holds up to a shift, and it is measured by
a weighted related entropy, for which we choose an appropriate entropy
associated with the strictly convex flux. In particular, we handle shocks with
small amplitude. This result improves the recent article [18] of the author and
Vasseur on -contraction property of shocks to scalar viscous conservation
laws with a special flux, that is almost the Burgers flux
Propagation of the mono-kinetic solution in the Cucker-Smale-type kinetic equations
In this paper, we study the propagation of the mono-kinetic distribution in
the Cucker-Smale-type kinetic equations. More precisely, if the initial
distribution is a Dirac mass for the variables other than the spatial variable,
then we prove that this "mono-kinetic" structure propagates in time. For that,
we first obtain the stability estimate of measure-valued solutions to the
kinetic equation, by which we ensure the uniqueness of the mono-kinetic
solution in the class of measure-valued solutions with compact supports. We
then show that the mono-kinetic distribution is a special measure-valued
solution. The uniqueness of the measure-valued solution implies the desired
propagation of mono-kinetic structure.Comment: 9 page
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