Delta shock waves in the shallow water system
The authors continue studies by \textit{C. O. R. Sarrico} [J. Math. Anal. Appl. 317, No. 2, 496--509 (2006; Zbl 1099.35121)], where gas dynamics phenomena, known as ``infinitely narrow soliton solutions'', discovered by \textit{V. P. Maslov} and \textit{G. A. Omel'yanov} [Russ. Math. Surv. 36, No. 3, 73--149 (1981; Zbl 0494.35080)], \textit{V. P. Maslov} and \textit{V. A. Tsupin} [Sov. Phys., Dokl. 24, 344--356 (1979; Zbl 0437.76064); translation from Dokl. Akad. Nauk SSSR 246, 298--300 (1979)], were used directly in the distributional form. This paper focuses on a Riemann problem for the shallow water system \[ u_t+ (v+\frac{1}{2}u^2 )_x=0, \] \[ v_t+ (u+ uv)_x =0 \] in the Sobolev-Schwartz space of generalized functions. Such a system of nonlinear conservation laws is a simplified model for the study of surface waves that are long when compared with the fluid's depth. The authors look for the solution as sum of the regular component and the singular component containing the Dirac delta functional. The concept of \(\alpha\)-solutions for the Riemann problem is introduced and evaluated. Finally, the physical nature of solutions is discussed.
scientific article; zbMATH DE number 6973933
scientific article