Delta-shock wave type solution of hyperbolic systems of conservation laws

Abstract

For two classes of hyperbolic systems of conservation laws <italic>new definitions of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shock wave type solution</italic> are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is <italic>relevant</italic> to the notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shocks. The <italic>weak asymptotics method</italic> developed by the authors is used to describe the propagation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shock waves to the three types of systems of conservation laws and derive the corresponding Rankine–Hugoniot conditions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shocks.