Convergence of viscosity solutions for \(2\times 2\) hyperbolic conservation laws with one characteristic field linearly degenerate on some zero measure sets
Consider the Cauchy problem for the general \(2\times 2\) hyperbolic conservation laws \[ u_t+ f(u, v)_x= 0,\quad v_t+ g(u, v)_x= 0,\tag{1} \] \[ (u, v)(x, 0)= (u_0(x), v_0(x)).\tag{2} \] Suppose that the two eigenvalues of system (1) are \(\lambda_1(u, v)\), \(\lambda_2(u, v)\), the corresponding Riemann invariants are \(w= w(u, t)\), \(z= z(u, v)\), and \(w= w(u, v)\), \(z= z(u, v)\) give a bijective smooth mapping from \((u, v)\) plane onto \((w, z)\) plane. We suppose that \(u_0(x)\), \(v_0(x)\) are bounded measurable functions, \(\lambda_1(u, v)< \lambda_2(u, v)\) and that one characteristic field of system (1) is genuinely nonlinear, and the other is linearly degenerate on some zero measure sets. In this note, we prove the convergence of viscosity solutions through analyzing the progressing entropy waves of Lax type, and we establish an existence theorem of the global weak solution for the Cauchy problem (1), (2).
scientific article; zbMATH DE number 883561