Related papers: Long-time behavior in scalar conservation laws
In this paper we present a novel framework for obtaining high order numerical methods for 1-D scalar conservation laws with non-convex flux functions. When solving Riemann problems, the Oleinik entropy condition, [16], is satisfied when the…
The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating…
We develop deterministic particle schemes to solve non-local scalar conservation laws with congestion. We show that the discrete approximations converge to the unique entropy solution with an explicit rate of convergence under more general…
We study the large-time behavior in all $L^p$ norms and in different space-time scales of solutions to a nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power of the Laplacian $(-\Delta)^s$, $s\in…
We consider well-balanced schemes for the following 1D scalar conservation law with source term: d_t u + d_x f(u) + z'(x) b(u) = 0. More precisely, we are interested in the numerical approximation of the initial boundary value problem for…
In this paper, we investigate the global structure of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscous/diffusive flux…
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic…
On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…
High-order accurate, $\textit{entropy stable}$ numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space…
We obtain a $C^1$-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin's entropy formula holds thus establishing the continuous-time version of \cite{T}. Moreover, in any compact manifold of…
We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the…
This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law…
We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator…
This paper is concerned with one-dimensional 2 x 2 systems of conservation laws with a flux f=f(x, U) that is discontinuous with respect to the spatial variable. No monotonicity assumption is imposed on the mapping x \to f(x,U). We…
We study a complex Ginzburg-Landau (GL) type model related to fluid instabilities in the boundary of magnetized toroidal plasmas (called edge-localized modes) with a prescribed shear flow on the Neumann boundary condition. We obtain the…
We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and…
In this paper, we are interested in an inverse problem for the active scalar equations with fractional dissipation on the torus. We perform a second order linearization to relate our model to the linear fractional diffusion equation. Our…
We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency…
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
We prove the asymptotic convergence of a space-periodic entropy solution of a one-dimensional degenerate parabolic equation to a traveling wave. It is also shown that on a segment containing the essential range of the limit profile the flux…