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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 3

### The generalized Riemann problem and instability of delta shock to the chromatography equations

Pages: 705 – 734

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n3.a5

#### Authors

#### Abstract

The generalized Riemann problem for the nonlinear chromatography equations in a neighborhood of the origin $(t \gt 0)$ on the $(x,t)$ plane is considered. The problem is quite different from the previous generalized Riemann problems which have no delta shock wave in the corresponding Riemann solutions. With the method of characteristic analysis and the local existence and uniqueness theorem proposed by Li Ta-tsien and Yu Wen-ci [T.T. Li and W.C. Yu, Duke Univ. Math. Ser. V, Durham, NC, 1985], we constructively solve the generalized Riemann problem and prove the existence and uniqueness of the solutions. It is proved that the generalized Riemann solutions possess a structure similar to the solution of the corresponding Riemann problem for most cases. In case that there is a delta shock wave in the corresponding Riemann solution, we discover that the generalized Riemann solution may turn into a combination of a shock wave and a contact discontinuity, which shows the instability and the internal mechanisms of a delta shock wave.

#### Keywords

chromatography equations, generalized Riemann problem, delta shock wave, instability, entropy condition

#### 2010 Mathematics Subject Classification

35L65, 35L67, 76L05, 76N10

Received 28 October 2016

Received revised 27 January 2018

Accepted 27 January 2018