Shock formation for 3D steady supersonic flows with general short pulse data

Abstract

This paper concerns the shock formation problem for the 3D steady supersonic potential equation of polytropic gases. The potential equation is described by a second order quasilinear wave equation $\sum _{i=1}^3[({\partial }_i\Phi {)}^2-c^2(\rho )]{\partial }_i^2\Phi +2\sum _{1\le i<j\le 3}{\partial }_i\Phi {\partial }_j\Phi {\partial }_{ij}^2\Phi =0$, where $x=(x^1,x^2,x^3)$, $({\partial }_1,{\partial }_2,{\partial }_3)=({\partial }_{x^1},{\partial }_{x^2},{\partial }_{x^3})$, $c(\rho )=\sqrt{p^{\prime }(\rho )}$ is the sonic speed with $p(\rho )=A{\rho }^{\gamma }$ ($\gamma >1$), and ${\partial }_3\Phi >c(\rho )$. For the short pulse boundary data $\Phi {|}_{x^3=0}={\delta }^{\nu }{\Phi }_0(\frac{r-1}{\delta },\omega )$ and ${\partial }_3\Phi {|}_{x^3=0}=q_0+{\delta }^{\nu -1}{\Phi }_1(\frac{r-1}{\delta },\omega )$ with $r=\sqrt{(x^1{)}^2+(x^2{)}^2}$, $\omega =(\frac{x^1}{r},\frac{x^2}{r})\in \mathbb{S}$, $1<\nu <2$ and small $\delta >0$, it is shown that a shock will be formed in a finite $x^3$-distance as long as the boundary data are supersonic and satisfy $({\Phi }_0,{\Phi }_1)\not\equiv 0$. This coincides with physical phenomenon that strong compression of supersonic polytropic gases yields shocks. One of our main ingredients is to find a good unknown so that the previously imposed compatibility conditions on the short pulse initial data are removed as well as the required weighted energy estimates in the existing literatures are derived. It is expected that the method here will be applied to study the shock formation problem with general short pulse initial data for the 3D steady supersonic Euler equations of polytropic gases.