Wave-front tracking for the equations of isentropic gas dynamics

Abstract

We study the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> × </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2 \times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> system of conservation laws of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v Subscript t Baseline minus u Subscript x Baseline equals u Subscript t Baseline plus p left-parenthesis v right-parenthesis Subscript x Baseline equals 0 comma p equals k squared v Superscript negative gamma Baseline left-parenthesis gamma greater-than-or-equal-to 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi> γ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> γ </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">v_t - u_x = u_t + p(v)_x =0, \ p = k^2v^{-\gamma } (\gamma \geq 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which are the model equations of isentropic gas dynamics. Weak global in time solutions are obtained by Nishida-Smoller (CPAM 1973) provided <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis gamma minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> γ </mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\gamma - 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> times the total variation of the initial data is sufficiently small. The aim of this paper is to give an alternative proof by using the Dafermos-Bressan-Risebro wave-front tracking scheme. We obtain new estimates of the total amount of interactions, which also imply the asymptotic decay of the solution. The main idea is to define appropriate amplitude to the path that is a continuation of shock fronts.