Formation of quasi-singularities of shock and implosion type for compressible Euler flows

Abstract

This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points $\{z_j{\}}_{j=1}^N\subset {\mathbb{R}}^3$ and an arbitrarily large parameter $M\gg 1$, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution $$(p,v)\in C^1([0,T]\times {\mathbb{R}}^3;\mathbb{R}\times {\mathbb{R}}^3),$$ with $$T=C_0{M}^{-\frac{3\gamma +1}{\gamma -1}},$$ where $\gamma >1$ is the adiabatic index and $C_0>0$ is a constant independent of $M$. Throughout the time interval $[0,T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point $z_j$. More precisely, each component of the velocity field is bounded below by $M$, each component of the pressure gradient is of size at least $M^{\frac{2\gamma }{\gamma -1}}$, and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least $M$ at every $z_j$. All implicit constants in these estimates are independent of $M$ and depend solely on the fixed physical parameters and the locations $\{z_j{\}}_{j=1}^N$. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.