Solutions of Hamilton–Jacobi Equations and Scalar Conservation Laws with Discontinuous Space–Time Dependence

Abstract

We establish a unique stable solution to the Hamilton-Jacobi equation u t þ HðKðx; tÞ; u x Þ ¼ 0;x 2 ðÀ1; 1Þ; t 2 ½0; 1Þwith Lipschitz initial condition, where Kðx; tÞ is allowed to be discontinuous in the ðx; tÞ plane along a finite number of (possibly intersecting) curves parameterized by t:We assume that for fixed k; Hðk; pÞ is convex in p and lim p!AE1 j H ðk;pÞ p j ¼ 1: The solution is determined by showing that if K is made smooth by convolving K in the x direction with the standard mollifier, then the control theory representation of the viscosity solution to the resulting Hamilton-Jacobi equation must converge uniformly as the mollification decreases to a Lipschitz continuous solution with an explicit control theory representation. This also defines the unique stable solution to the corresponding scalar conservation law u t þ ðf ðKðx; tÞ; uÞÞ x ¼ 0;x 2 ðÀ1; 1Þ; t 2 ½0; 1Þwith K discontinuous. # 2002 Elsevier Science (USA)

Convergence of the Lax-Friedrichs Scheme and Stability for Conservation Laws With a Discontinuous Space-Time Dependent Flux

“…After the main results of this paper were obtained, we learned of a preprint [5] in which the authors use the singular mapping approach to prove convergence of a front tracking scheme for (1.1) when the coefficient k(x, t) has a multiplicative space-time dependence a(x)g(t) > 0, the nonlinearity u → f (k, u) is concave, the mapping k → f (k, u) is nondecreasing, and the initial function u 0 (x) is roughly speaking of bounded total variation. We would like to stress that the existence result given herein holds under conditions that are significantly more general than those needed for the existence results in [27,5].…”

“…In addition, more or less for free, our method of analysis allows for a time dependent coefficient. The time dependent case was treated only recently in [27]. The author proved existence of a unique and stable solution under the assumption that u → f (k, u) is convex and k(x, t) is piecewise smooth, i.e., k(x, t) is allowed to be discontinuous along a finite number of curves in the (x, t) plane.…”

Unconditionally Stable Methods for Hamilton–Jacobi Equations

“…In particular, if H is convex, we know that the problem (23) has a unique solution-the so-called entropy solution-that is the limit of the corresponding regularized solutions. Furthermore, according to recent results of Ostrov [42], if p → H (a, p) is convex, we can define "viscosity solutions" of (22) even when a has a finite number of jump discontinuities. These are defined as the (unique!)…”

A Hamilton-Jacobi approach to junction problems and application to traffic flows

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