Convergence of viscosity solutions for 2 X 2 hyperbolic conservation laws with one characteristic field linearly degenerate on some zero measure sets

摘要

Consider the Cauchy problem for the generally 2 X 2 hyperbolic conservation laws. Suppose that the two eigenvalues of system (0.1) are λ_1(u, v), λ_2(u, v), the corresponding Riemann invariants are w=w(u, v), 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. Throughout this note, we always suppose that A_1 u_0(x), v_0(x) are bounded measurable functions. A_2 λ_1(u, v), λ_2(u, v)∈C~1 and system (0.1) are strictly hyperbolic, i.e. λ_1(u, v)<λ_2(u, v). A_3 One characteristic field of system (0.1) is genuinely nonlinear, the other is linearly degenerate on sortie zero measure sets. Without loss of generality, we can assume λ_(1z)(w, z)≠0and for any fixed z_0∈R.