New δ-shock waves in the p-system: a distributional product approach

This article studies a Riemann problem for the so-called “[Formula: see text]-system”[Formula: see text], [Formula: see text], which rules one-dimensional isentropic thermoelastic media. Such study is made using a product of distributions that allows us to extend both the classical solution concept and a weak solution concept. By considering [Formula: see text] as an entire function that takes real values on the real axis, this product also extends for certain distributions [Formula: see text] the meaning of [Formula: see text]. Under certain conditions, this Riemann problem has solutions that are [Formula: see text]-shock waves. Furthermore, those [Formula: see text]-shock waves satisfy the so-called generalized Rankine–Hugoniot conditions.

Distributions as initial values in a triangular hyperbolic system of conservation laws

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

New Distributional Global Solutions for the Hunter-Saxton Equation

In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation[ut+((1/2)u2)x]x=(1/2)ux2in a convenient space of discontinuous functions. With the help of a consistent extension of the classical solution concept, two classes of discontinuous solutions are obtained: one class of conservative solutions and another of dispersive solutions. A necessary and sufficient condition for the propagation of a distributional profile as a travelling wave is also presented, which allows identifying an interesting set of explicit distributional travelling waves. In the paper, we will show some results we have obtained by applying this framework to other equations and systems.

A DISTRIBUTIONAL PRODUCT APPROACH TO δ-SHOCK WAVE SOLUTIONS FOR A GENERALIZED PRESSURELESS GAS DYNAMICS SYSTEM

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show the possible arising of a δ-shock wave solution. This procedure affords a simpler and more general framework to construct singular solutions and can surely be applied to other equations or systems.