Formation and Construction of a Multidimensional Shock Wave for the First-Order Hyperbolic Conservation Law with Smooth Initial Data

In this paper, the problem on formation and construction of a multidimensional shock wave is studied for the first order conservation law ∂tu+∂xF (u)+∂yG(u) = 0 with smooth initial data u0(x, y). It is wellknown that the smooth solution u will blow up on the time T ∗ = − 1 minH(ξ,η) when minH(ξ, η) < 0 holds for H(ξ, η) = ∂ξ(F (u0(ξ, η))) + ∂η(G (u0(ξ, η))), more precisely, only the first order derivatives ∇t,x,yu blow up on t = T ∗ meanwhile u itself is still continuous until t = T ∗. Under the generic nondegenerate condition of H(ξ, η), we construct a local weak entropy solution u for t ≥ T ∗ which is not uniformly Lipschitz continuous on two sides of a shock surface Σ. The strength of the constructed shock is zero on the initial blowup curve Γ and then gradually increases for t > T ∗. Additionally, in the neighbourhood of Γ, some detailed and precise descriptions on the singularities of solution u are given.