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Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

Feng Shao, Dongyi Wei, Shumao Wang, Zhifei Zhang

TL;DR

The paper resolves the degenerate case $\gamma=5/3$ for the 3-D isentropic compressible Navier–Stokes equations by combining a construction of smooth self-similar Euler imploding profiles with a nonlinear framework (CLGSSS) to produce non-radial Navier–Stokes blow-up data. It first proves the existence of a discrete sequence of smooth radially symmetric self-similar Euler profiles $(\mathbf U_E,S_E)$ corresponding to exponents $r_n$ with $r_n<3-\sqrt{3}$ and $r_n\to3-\sqrt{3}$, and then extends these to finite-time blow-up solutions of the 3-D Navier–Stokes system with $\nu=1$ and non-radial perturbations, via a coercivity/repulsivity analysis of an autonomous ODE system obtained from the self-similar reduction. A computer-assisted verification confirms a key non-degeneracy quantity $\mathcal S_\infty>0$, enabling the construction, and yielding explicit asymptotics for the blow-up profiles. The results advance the Merle–Raphaël–Rodnianski–Szeftel program to the physically important monatomic gas case, establishing the existence of infinitely many smooth blow-up solutions in 3-D with self-similar structure for $\gamma=5/3$ and providing new insight into front-compression mechanisms in compressible fluids.

Abstract

In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $γ=5/3$, which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Raphaël, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for $γ=5/3$. Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.

Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

TL;DR

The paper resolves the degenerate case for the 3-D isentropic compressible Navier–Stokes equations by combining a construction of smooth self-similar Euler imploding profiles with a nonlinear framework (CLGSSS) to produce non-radial Navier–Stokes blow-up data. It first proves the existence of a discrete sequence of smooth radially symmetric self-similar Euler profiles corresponding to exponents with and , and then extends these to finite-time blow-up solutions of the 3-D Navier–Stokes system with and non-radial perturbations, via a coercivity/repulsivity analysis of an autonomous ODE system obtained from the self-similar reduction. A computer-assisted verification confirms a key non-degeneracy quantity , enabling the construction, and yielding explicit asymptotics for the blow-up profiles. The results advance the Merle–Raphaël–Rodnianski–Szeftel program to the physically important monatomic gas case, establishing the existence of infinitely many smooth blow-up solutions in 3-D with self-similar structure for and providing new insight into front-compression mechanisms in compressible fluids.

Abstract

In this paper, we prove the blow-up of the -D isentropic compressible Navier-Stokes equations for the adiabatic exponent , which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Raphaël, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for . Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.
Paper Structure (26 sections, 43 theorems, 419 equations, 4 figures)

This paper contains 26 sections, 43 theorems, 419 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Compressible Euler
  4. Compressible Navier-Stokes
  5. Self-similar solutions
  6. Main results
  7. The autonomous ODE system and sketch of the proof
  8. The autonomous ODE system and phase portraits
  9. Sketch of the proof: existence
  10. Sketch of the proof: repulsivity
  11. Local smooth solution around the sonic point
  12. Fundamental properties of the phase portrait near $Q_2$
  13. Analytic solutions near $Q_2$
  14. Quantitative properties of local analytic solutions
  15. Reformulation of the recurrence relation
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $d=3$ and $\gamma=5/3$. There exists a discrete sequence $\{r_n\}$ satisfying such that profile_Euler with $r=r_n$ admits a global $C^\infty$ radially symmetric solution $(\mathbf U_E, S_E)$ possessing the following properties: for some constant $\eta>0$ depending on $r$, recalling Eq.radial_transform.

Figures (4)

  • Figure 1: Phase portrait for the $\sigma-w$ system \ref{['autonomousODE']}: Dashed curve is the trajectory of the solution constructed in Theorem \ref{['Thm.ODE']}.
  • Figure 2: Phase portrait for the $\tau-u$ system \ref{['ODE_u_tau']}: Dashed curve is the trajectory of the solution.
  • Figure 3: Phase portrait for the $\sigma-w$ system \ref{['ODE_w_sigma']}: Positions of auxiliary points $P_A$ and $P_B$.
  • Figure 4: Phase portrait for the $\tau-u$ system \ref{['ODE_u_tau']}: Positions of auxiliary points $Q_A$ and $Q_B$.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2: CLGSSS
  • Corollary 1.3
  • Theorem 2.1: Existence
  • Proposition 2.2: Repulsivity
  • Remark 2.1
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 75 more