Related papers: Scalar conservation laws with multiple rough fluxe…
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,…
A quasi-linear hyperbolic partial differential equation with a discontinuous flux models geologic carbon dioxide migration and storage. Dual flux curves characterize the model, giving rise to flux discontinuities. One convex flux describes…
We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model…
We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the…
In studies of interfaces with dynamic chemical composition, bulk and interfacial quantities coupled via surface conservation laws of excess surface quantities. While this approach is for microscopically sharp interfaces, its applicability…
We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions…
Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We introduce a novel stochastic perturbation of scalar…
We prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non-autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.
We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following the work of Bouchut and James (Comm. Partial Diff. Eq., 24, 1999), a notion of duality solution for…
The notes are an overview of part of the theory of pathwise weak solutions to two classes of scalar fully nonlinear first- and second-order degenerate parabolic partial differential equations with multiplicative rough time dependence, a…
We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle…
We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution…
We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow, without any additional conditions on finiteness/discreteness of the set of discontinuities or on the monotonicity of…
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…
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 consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L^2 perturbations of shock wave solutions to the Riemann problem using the relative entropy…
We consider the problem of existence of entropy weak solutions to scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect both to the space variable x and the unknown quantity u. The problem…
Scalar conservation laws in one space variable allow a Lagrangian (particle path) formulation. The Lagrangian trajectory in the infinite-dimensional group of diffeomorphisms on the physical space can be written as a system of conservation…
The entropy based flux-limiting (EFL) scheme is a novel approach designed to accurately resolve shocks and discontinuities in special and general relativistic hydrodynamics. By adaptively adjusting the numerical fluxes, the EFL method…
In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique…