Newton’s second law and the multiplication of distributions

Sarrico, C. O. R.; Paiva, A.

doi:10.1063/1.5021949

Cited by 12 publications

(6 citation statements)

References 20 publications

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“…In this direction, we also mention the -product by Sarricco (see e.g. [ 51 , 52 , 57 , 58 ] and references therein) which is applied in the case of various systems of conservation laws and similar equations and which works in the case of fully nonlinear equation (i.e. nonlinear with respect to all unknowns).…”

Section: Introductionmentioning

confidence: 99%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any $$n\times n$$ n × n system of conservation laws has a solution. The solution concept is an extension of the notion of singular $$\delta $$ δ -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of $$\delta $$ δ -distributions as a part of solutions to systems of conservation laws.

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Section: Introductionmentioning

confidence: 99%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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“…So we may visualize the distinction between the delta shock wave to the ordinary shock wave as more characteristics enter the delta shock wave than the latter. Diverse methods can be used to explore system of hyperbolic conservation laws in the context of Riemann problem such as vanishing pressure limit method, [14][15][16][17][18] weak asymptotic method, 19 distributional product method, 20,21 flux approximation method, [22][23][24][25] split delta function method, 26,27 etc. In Joseph and Sahoo, 28 vanishing viscosity method is used for a system consisting of four conservation laws.…”

Section: Introductionmentioning

confidence: 99%

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

Chhatria

Sen

Sekhar

2022

Math Methods in App Sciences

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Here, we study the Riemann problem for a strictly hyperbolic system of conservation laws, which occurs in gas dynamics and nonlinear elasticity. We establish the existence and uniqueness of the solution of Riemann problem containing delta shock wave by employing self-similar vanishing viscosity approach. We prove that delta shock is stable under self-similar viscosity perturbation, which ensures that delta shock wave is a unique entropy solution.

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“…It may even happen that those weak limits cannot be substituted into equations or systems owing to the well known difficulties of multiplying distributions. On top of that, limit processes involving sequences of continuous functions may not yield to mathematically consistent solutions (see [13], Section II). Our method overcome those difficulties, as we will explain.…”

Section: Introduction and Contentsmentioning

confidence: 99%

Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions

Sarrico¹

2021

JNMP

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The present paper concerns the study of a Riemann problem for the conservation lawwhere x, t, k, v and u = u(x,t) are real numbers. We consider φ an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions we will explicitly see the possible emergence of waves with the shape of shock waves, delta waves and delta shock waves. For this purpose, we define a rigorous concept of a solution which extends both the classical solution concept and a weak solution concept. All this framework is developed in the setting of a distributional product that is not constructed by approximation. We include the main ideas of this product for the reader's convenience. Recall that delta shock waves are relevant physical phenomena which may be interpreted as processes of concentration of mass or even as processes of formation of galaxies in the universe.

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Newton’s second law and the multiplication of distributions

Cited by 12 publications

References 20 publications

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions

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