相关论文: Wave-front tracking for a quasi-linear scalar cons…

We consider the Cauchy problem for the quasi-linear scalar conservation law \[u_t+\mathcal{F}(u)_t+u_x=0,\] where $\mathcal{F}$ is a specific hysteresis operator. Hysteresis models a rate-independent memory relationship between the input…

This article studies the Cauchy problem for the scalar conservation law \[ \partial_t u + \partial_t w + \partial_x f(u) = 0, \] where $w(x,t) = [\mathcal{F}(u)(x,t)]$ is the output of a specific hysteresis operator, namely the Play…

In this article, we propose a novel front tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and prove that approximations converge to the unique entropy solution. The main tools are Dafermos'…

This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in $2\times 2$ hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed…

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative,…

We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise…

We study in what sense one can determine the function $k=k(x)$ in the scalar hyperbolic conservation law $u_t+(k(x)f(u))_x=0$ by observing the solution $u(t,\dott)$ of the Cauchy problem with initial data $u|_{t=0}=u_o$.

In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when…

We deal with the Cauchy problem for multi-dimensional scalar conservation laws, where the fluxes and the source terms can be discontinuous functions of the unknown. The main novelty of the paper is the introduction of a~kinetic formulation…

In this paper, we investigate the asymptotic behavior 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 viscosity is of…

We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…

A problem of a wave identification is formulated. An example is considered in conditions of one-dimensional Cauchy problem for conventional string equation in matrix form and its inhomogeneous two-component version. The acoustic and…

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a non-local constraint. Existence and stability results for the Cauchy problem with Lipschitz…

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative,…

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by L\'evy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of…

Consider a scalar conservation law with discontinuous flux \begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \end{equation*} where $u=u(x,t)$ is…

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous…

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of…

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-i\theta}u_t&=&u_{xx}-|u|^2u-\alpha g(v)u& v_t+(f(v))_x&=&\alpha (g'(v)|u|^2)_x&…