On the behavior of the boundary of a movable volume in a smooth flow of a compressible fluid (Q2480454)

The author considers the following system of conservation laws in differential form: \[ \rho(\partial_t\vec V+ (\vec V,\nabla_x)\vec V)= -\nabla_x P,\quad x\in\mathbb{R}^n,\tag{1} \] \[ \partial_t\rho+ \text{div}(\rho\vec V)= 0,\tag{2} \] \[ \partial_t S+(\vec V,\nabla_x)S= 0,\tag{3} \] where the density, velocity vector, and entropy are the unknowns and are denoted by \(\rho\), \(\vec V=(V_1,\dots, V_n)\), and \(S\) respectively. Here \(P(t,x)\) is the pressure, and the adiabatic exponent is denoted by \(\gamma\) \((\gamma> 1)\), that is \(P(t,x)= \rho^\gamma e^S\). The author addresses the following question: Consider a finite movable volume \(W(t)\) of a compressible fluid whose motion is described by system (1)--(3). Suppose that the point \(x_0\) does belong to the volume \(W(t)\) at the initial time. What conditions should be imposed on the initial data, so as to ensure that the boundary of the \(\partial W(t)\) reaches a given \(\varepsilon\)-neighborhood of the point \(x_0\) within time in which the flow preserves its smoothness.