We establish the equivalence of free piston and delta shock, for the one-space-dimensional pressureless compressible Euler equations. The delta shock appearing in the singular Riemann problem is exactly the piston that may move freely forward or backward in a straight tube, driven by the pressureless Euler flows on two sides of it in the tube. This result not only helps to understand the physics of the somewhat mysterious delta shocks, but also provides a way to reduce the fluid–solid interaction problem, which consists of several initial-boundary value problems coupled with moving boundaries, to a simpler Cauchy problem. We show the equivalence from three different perspectives. The first one is from the sticky particles, and derives the ordinary differential equation (ODE) of the trajectory of the piston by a straightforward application of conservation law of momentum, which is physically simple and clear. The second one is to study a coupled initial-boundary value problem of pressureless Euler equations, with the piston as a moving boundary following the Newton’s second law. It depends on a concept of Radon measure solutions of initial-boundary value problems of the compressible Euler equations which enables us to calculate the force on the piston given by the flow. The last one is to solve directly the singular Riemann problem and obtain the ODE of delta shock by the generalized Rankine–Hugoniot conditions. All the three methods lead to the same ODE. 

/pdf/delta-shock-as-free-piston-in-pressureless-euler-flows-204j9t7v.pdf

Delta shock as free piston in pressureless Euler flows

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions. 

/pdf/radon-measure-solutions-to-riemann-problems-for-isentropic-1qoujxlx.pdf

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law

Pseudo-Index Theory for a Schrödinger Equation with Competing Potentials

pdf/hypersonic-similarity-for-steady-compressible-full-euler-1svu4f7kh6.pdf

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

The Multiplicity and Concentration of Positive Solutions for the Kirchhoff-Choquard Equation with Magnetic Fields

Radon measure solutions for steady hypersonic-limit Euler flows passing two-dimensional finite non-symmetric obstacles and interactions of free concentration layers

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The Multiplicity and Concentration of Positive Solutions for the Kirchhoff-Choquard Equation with Magnetic Fields

Radon measure solutions for steady hypersonic-limit Euler flows passing two-dimensional finite non-symmetric obstacles and interactions of free concentration layers