[Paper Review] Formation of shocks for 2D isentropic compressible Euler

Tristan Buckmaster, Steve Shkoller+1 more|arXiv (Cornell University)|Jul 8, 2019

Navier-Stokes equation solutions39 references22 citations

TL;DR

This paper constructs smooth, finite-energy solutions to the 2D isentropic compressible Euler equations with $̲\gamma > 1$ that form shocks in finite time $\mathcal{O}(\varepsilon)$ from initial data with $\mathcal{O}(1)$ amplitude and a minimum slope of $-1/\varepsilon$. Using a self-similar transformation and modulated variables, it proves the existence of cusp-type shocks with $C^{1/3}$ Hölder regularity and nontrivial vorticity, avoiding perturbations from irrotational flows by focusing on purely azimuthal wave dynamics.

ABSTRACT

We consider the 2D isentropic compressible Euler equations, with pressure law $p(ρ) = (\sfrac{1}γ) ρ^γ$, with $γ>1$. We provide an elementary constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, and with {nontrivial vorticity}. We prove that for initial data which has minimum slope $- {\sfrac{1}{ \eps}}$, for $ \eps>0$ taken sufficiently small relative to the $\OO(1)$ amplitude, there exist smooth solutions to the Euler equations which form a shock in time $\OO(\eps)$. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with Hölder $C^ {\sfrac{1}{3}}$ regularity. Our objective is the construction of solutions with inherent $\OO(1)$ vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann-type variables to obtain a system of forced transport equations. Using a transformation to modulated self-similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self-similar time, of a smooth blowup profile.

Motivation & Objective

• To construct smooth initial data with $\mathcal{O}(1)$ amplitude and a minimum slope of $-1/\varepsilon$ that lead to shock formation in finite time.
• To demonstrate shock formation in the 2D isentropic compressible Euler equations with nontrivial vorticity, avoiding small perturbations of irrotational flows.
• To explicitly compute the blowup time $T_* = \mathcal{O}(\varepsilon)$ and location for solutions dominated by purely azimuthal wave motion.
• To establish global stability in self-similar time of a smooth blowup profile using modulated self-similar variables and pointwise estimates.
• To characterize the shock profile at $t = T_*$ as a cusp with $C^{1/3}$ regularity, even in the presence of $\mathcal{O}(1)$ vorticity.

Proposed method

• Use of homogeneous solutions to the Euler equations to reduce dynamics to a regime dominated by azimuthal wave motion.
• Transformation to modulated self-similar variables to track blowup location, time, and amplitude dynamically.
• Derivation of a system of forced transport equations in Riemann-type variables for the perturbed azimuthal wave components.
• Application of pointwise estimates to prove global stability of the blowup profile in self-similar time.
• Use of a damping and forcing framework in Lemma A.2 to control solution growth and prevent blowup in the outer region.
• Explicit computation of blowup time and location via the self-similar transformation and modulation functions.

Experimental results

Research questions

• RQ1 Can shock formation be constructed in the 2D isentropic compressible Euler equations with $\mathcal{O}(1)$ vorticity and non-irrotational initial data?
• RQ2 What is the precise blowup time and location for solutions with initial slope $-1/\varepsilon$ and $\mathcal{O}(1)$ amplitude?
• RQ3 What is the regularity of the shock profile at the blowup time, and can it be explicitly characterized?
• RQ4 How can a self-similar framework with modulation functions stabilize the blowup profile globally in self-similar time?
• RQ5 Does the dynamics remain dominated by azimuthal waves, with bounded radial components, during shock formation?

Key findings

• Solutions with initial slope $-1/\varepsilon$ form a shock in time $T_* = \mathcal{O}(\varepsilon)$, with blowup time and location explicitly computable.
• The shock profile at $t = T_*$ exhibits $C^{1/3}$ Hölder regularity, forming a cusp-like singularity.
• The solutions maintain $\mathcal{O}(1)$ vorticity at the shock, distinguishing them from irrotational shock constructions.
• The blowup is stable in self-similar time, with the profile globally stable under the modulated self-similar transformation.
• For $\gamma = 3$, the dynamics reduce to an elementary study of the Burgers equation, fully characterizing shock formation.
• No other singularity forms before $T_*$, and $|\nabla \rho|$ and $|\nabla u|$ both blow up as $t \to T_*$.