A Complete Global Solution to the Pressure Gradient Equation
Zhen Lei, Yuxi Zheng
TL;DR
This paper investigates the global existence of solutions to a two-dimensional pressure gradient equation Riemann problem, demonstrating that the vacuum boundary coincides with the coordinate axes, extending previous results on smooth solutions.
Contribution
It proves that the vacuum boundary in the pressure gradient equation Riemann problem is the trivial coordinate axes, providing a complete global solution analysis.
Findings
Vacuum boundary is the coordinate axes.
Global smooth solutions exist up to the vacuum boundary.
Extension of previous results on pressure gradient equations.
Abstract
We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants. The global existence of a smooth solution was established in Dai and Zhang [Arch. Rational Mech. Anal., {\bf 155}(2000), 277-298] up to the free boundary of vacuum. We prove that the vacuum boundary where the system is degenerate is the trivial coordinate axes.
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Taxonomy
TopicsNavier-Stokes equation solutions · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations