Multiplication of distributions and travelling wave solutions for the Keyfitz-Kranzer system
type: Item
instance of: scholarly article
title: Multiplication of distributions and travelling wave solutions for the Keyfitz-Kranzer system
zbMATH Open document ID: 1412.46049
published in: Taiwanese Journal of Mathematics
publication date: 2018-10-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.twjm/1513393252
review text: Since the impossibility result on the multiplication of distributions, it is well known that the space of distributions is not suitable for studying nonlinear operations, in particular, nonlinear partial differential equations. Nevertheless, certain authors have introduced an intrinsic multiplication of distributions which works for some classes of distributions. The author of this paper first recalls the definition of an intrinsic multiplication called \(\alpha\)-product of distributions depending on a regularizing test function \(\alpha\) satisfying \(\int \alpha =1,\) and gives some properties of this product. Rephrasing the following system of nonlinear partial differential equations \[ \begin{aligned} u_{t}+(u^{2}-v)_{x}&=0, \\ v_{t}+(u^{3}/3-u)_{x}&=0,\end{aligned} \] on the basis of his definition, he introduces the concept of \(\alpha\)-solutions for the system. A~comparison with classical solutions is given. In the last section, travelling wave solutions with distributional profiles for the cited system are studied as well as necessary and sufficient conditions for the propagation of distributional wave profiles.
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 35D99
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35F50
zbMATH DE Number: 6965392
zbMATH Keywords: product of distributions
zbMATH Keywords: conservation laws
zbMATH Keywords: travelling shock waves
zbMATH Keywords: travelling delta waves
zbMATH Keywords: travelling waves which are not measures
zbMATH Keywords: Keyfitz-Kranzer system
author: Carlos Orlando R. Sarrico
reviewed by: Chikh Bouzar
MaRDI profile type: Publication
OpenAlex ID: W2773882366
cites work: Q3799227
cites work: Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations
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 of a fully nonlinear 2 × 2 system of conservation laws
cites work: Q4263255
cites work: Singular shocks: retrospective and prospective
cites work: The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy
cites work: Spaces of weighted measures for conservation laws with singular shock solutions
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: 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: Nonstrictly hyperbolic conservation laws with a parabolic line
cites work: Existence of Dafermos profiles for singular shocks
cites work: Q5528072
cites work: Q4861428
DOI: 10.11650/TJM/171104
description: scientific article; zbMATH DE number 6965392