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

✍ Scribed by Daniel N. Ostrov

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 235 KB
• Volume: 182
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.2001.4088

✦ Synopsis

We establish a unique stable solution to the Hamilton-Jacobi equation

x 2 ðÀ1; 1Þ; t 2 ½0; 1Þ

with 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; 1Þ with K discontinuous.