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We describe a new method which allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a…
We prove that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of $\Delta x^2$ in the 1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that the…
In this article, we consider scalar conservation laws with fluxes having spatial discontinuities and possible flat regions and study the following three aspects: (i) existence, (ii) uniqueness and (iii) BV regularity of solutions. We…
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $\mu$…
We prove the local existence and uniqueness of solutions to a system of quasi-linear wave equations involving a jump discontinuity in the lower order terms. A continuation principle is also established.
The aim of this work is to make a further step towards the understanding of the near-critical reflection of internal gravity waves from a slope in the more general and realistic context where the size of viscosity $\nu$ and the size of…
We propose a spectral method based on the implementation of Chebyshev polynomials to study a model of conservation laws on network. We avoid the Gibbs phenomenon near shock discontinuities by implementing a filter in the frequency space in…
We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a…
To describe the small-scale intermittency of turbulence, a self-similarity is assumed for the probability density function of a logarithm of the rate of energy dissipation smoothed over a length scale among those in the inertial range. The…
In this paper, we establish a small time large deviation principles for scalar stochastic conservation laws driven by multiplicative noise. The doubling of variables method plays a key role.
We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone…
We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t$ is $BV$-regular and may exhibit…
In the case of scalar conservation laws $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were…
Method of compensated compactness is used to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it Bardos et.al}\cite{MR542510}) of the corresponding scalar…
Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux…
Spherical and cylindrical KdV-B equations have few known exact solutions, yet these solutions are hard to be interpreted physically. But these equations do have a family of diverging shock waves. Their properties such as asymptotic modes,…
In this paper we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with scale-invariant damping. We obtain estimates for the solution and its energy with the same decay rate of the…
We use Velocity Averaging lemma to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it F. Otto}) of the corresponding scalar conservation laws on a bounded domain…
A new kinetic model is proposed where the equilibrium distribution with bounded support has a range of velocities about two average velocities in 1D. In 2D, the equilibrium distribution function has a range of velocities about four average…
This work is a follow-up on [GOVW]. In that previous work a two-scale approach was used to prove the logarithmic Sobolev inequality for a system of spins with fixed mean whose potential is a bounded perturbation of a Gaussian, and to derive…