Formation of quasi-singularities of shock and implosion type for compressible Euler flows
This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in $\mathbb{R}^d$ for $d = 2, 3$. For any prescribed finite set of points and any sufficiently large parameter $\mathcal{M} > 0$, we construct a family of smooth, even analytic initial data whose corresponding solutions exhibit three concurrent properties. First, for each datum, there exists a time $T>0$ for which the corresponding solution remains $C^{1}$-smooth on $[0,T]$. Second, throughout this interval, the velocity and pressure gradient exceed $\mathcal{M}$, while the velocity gradient exceeds $\mathcal{M}^{2}$, in a small neighborhood of each prescribed point. Third, in sharp contrast, the velocity, its gradient, and the pressure gradient remain uniformly bounded -- independent of $\mathcal{M}$ -- in a fixed interior region whose boundary contains the designated points. Furthermore, the set of almost blowup points, namely points where the above amplification occurs, has vanishing measure and concentrates around the designated locations as $\mathcal M\to\infty$. This phenomenon generates highly localized quasi-singular structures exhibiting arbitrarily strong singular behavior, characterized by shock-like gradient concentration and implosion-like spatial localization, while remaining within the class of smooth solutions. The construction is based on specially designed profile to linearized compressible Euler equations, together with quantitative estimates and control of the underlying quasilinear hyperbolic system.