Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence
A unique stable solution \(u(x,t)\) to Hamilton-Jacobi equations of the form \[ u_t+ H(K(x, t), u_x)= 0,\quad x\in (-\infty,\infty),\quad t\in [0,\infty),\quad u(x,0)= g(x), \] where \(H\) is convex in \(u_x\) and has superlinear growth as \(u_x\to \pm\infty\), \(g(x)\) is Lipschitz continuous, and \(K\) is discontinuous along a finite number of curves in the \((x,t)\) plane, is constructed. 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 Hamiltonian-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 \(u_t+ (f(K(x, t), u))_x= 0\), \(x\in (-\infty,\infty)\), \(t\in [0,\infty)\) with \(K\) discontinuous.
scientific article; zbMATH DE number 1797553
scientific article