Global non-selfsimilar Riemann solutions for multi-dimensional scalar conservation laws with time-dependent nonlinear source terms

Abstract

In this paper, we are concerned with the global non-selfsimilar Riemann solutions for the multi-dimensional scalar conservation law with a time-dependent nonlinear source term 𝑔 ⁡ ( 𝑡 , 𝑢 ) , where the Riemann-type initial data consist of two constant states separated by a curved initial discontinuity, namely an ( 𝑛 − 1 ) -dimensional smooth manifold. For this Riemann problem, under the Lipschitz relating condition on the source term 𝑔 ⁡ ( 𝑡 , 𝑢 ) in (8) and Condition ℋ in (9) , we construct global the non-selfsimilar Riemann solutions, including non-selfsimilar shock waves and non-selfsimilar rarefaction waves, in which the shock surface and the solution in the rarefaction region are expressed by implicit functions, respectively. Furthermore, we prove the uniqueness by showing that these global non-selfsimilar Riemann solutions are Kruzkov entropy solutions. Finally, we present two examples to illustrate the global structures of multi-dimensional non-selfsimilar shock waves and rarefaction waves.