Delta shock waves in the shallow water system
type: Item
instance of: scholarly article
title: Delta shock waves in the shallow water system
zbMATH Open document ID: 1411.46034
author: Yanyan Li
published in: Journal of Dynamics and Differential Equations
publication date: 2018-11-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 35D99
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 6973933
zbMATH Keywords: products of distributions
zbMATH Keywords: shallow water system
zbMATH Keywords: shock waves
zbMATH Keywords: delta waves
zbMATH Keywords: delta shock waves
reviewed by: Denis Sidorov
MaRDI profile type: Publication
full work available at URL: s10884-017-9594-2
OpenAlex ID: W2611194441
cites work: Energy balance for undular bores
cites work: Q3799227
cites work: Multiplications of distributions in elasticity and hydrodynamics
cites work: Definition and weak stability of nonconservative products
cites work: Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems
cites work: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws
cites work: A contribution to the theory of generalized functions
cites work: Measure solutions to a strictly hyperbolic system of conservation laws
cites work: Singular solutions for the shallow-water equations
cites work: Singular solutions of a fully nonlinear 2 × 2 system of conservation laws
cites work: Non-standard characteristics in asymptotic problems
cites work: Asymptotic soliton-form solutions of equations with small dispersion
cites work: Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process
cites work: Q3814225
cites work: Distributional products and global solutions for nonconservative inviscid Burgers equation
cites work: New solutions for the one-dimensional nonconservative inviscid Burgers equation
cites work: Collision of delta-waves in a turbulent model studied via a distribution product
cites work: Products of distributions and singular travelling waves as solutions of advection-reaction equations
cites work: Products of distributions, conservation laws and the propagation of \(\delta'\)-shock waves
cites work: Q4921963
cites work: A DISTRIBUTIONAL PRODUCT APPROACH TO δ-SHOCK WAVE SOLUTIONS FOR A GENERALIZED PRESSURELESS GAS DYNAMICS SYSTEM
cites work: The Brio system with initial conditions involving Dirac masses: a result afforded by a distributional product
cites work: New distributional global solutions for the Hunter-Saxton equation
cites work: The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product
cites work: Products of distributions and collision of a δ-wave with a δ′-wave in a turbulent model
cites work: Q4943599
DOI: 10.1007/S10884-017-9594-2
Recommended article: Delta shock waves in shallow water flow
Recommended article: Delta shocks interactions and the wave front tracking method
Recommended article: Shallow Water Wave Systems
Recommended article: DELTA SHOCK WAVES AS A LIMIT OF SHOCK WAVES
Recommended article: Q4787062
Recommended article: Breaking of a weak shock wave on shallow water
Recommended article: Q3832247
Recommended article: Q3496066
Recommended article: Water waves in shallow channels of rapidly varying depth
description: scientific article; zbMATH DE number 6973933