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Development of Implosions of Solutions to the Three-Dimensional Degenerate Compressible Navier-Stokes Equations

Gui-Qiang G. Chen, Lihui Liu, Shengguo Zhu

Abstract

A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show that there exist smooth initial data for which the corresponding smooth solutions of the barotropic flow undergo finite-time implosion at the origin, with the density blowing up to infinity. In contrast, when the viscosity coefficients depend linearly on the density (as in the shallow water case), it has been established that, for general large spherically symmetric initial data, the solutions remain globally regular. These results indicate that the qualitative behavior of multidimensional solutions is sensitive to the structure of the viscosity coefficients. In this paper, we investigate the case of nonlinear viscosity coefficients with power-law density dependence. We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data with strictly positive density for which the corresponding smooth solutions implode in finite time at the origin. The key issue is to show that, in this regime, the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism driving the implosion. Establishing this rigorously is highly nontrivial due to the degenerate structure of the Navier-Stokes equations. To overcome this difficulty, we first derive a pointwise estimate for the density and then obtain spatial decay estimates for the velocity gradient via carefully constructed weighted high-order energy estimates and interpolation inequalities. The resulting decay rate is sufficiently rapid to compensate for the singular growth of the density, leading to uniform-in-time control of the viscous terms and ultimately to the formation of implosion.

Development of Implosions of Solutions to the Three-Dimensional Degenerate Compressible Navier-Stokes Equations

Abstract

A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show that there exist smooth initial data for which the corresponding smooth solutions of the barotropic flow undergo finite-time implosion at the origin, with the density blowing up to infinity. In contrast, when the viscosity coefficients depend linearly on the density (as in the shallow water case), it has been established that, for general large spherically symmetric initial data, the solutions remain globally regular. These results indicate that the qualitative behavior of multidimensional solutions is sensitive to the structure of the viscosity coefficients. In this paper, we investigate the case of nonlinear viscosity coefficients with power-law density dependence. We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data with strictly positive density for which the corresponding smooth solutions implode in finite time at the origin. The key issue is to show that, in this regime, the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism driving the implosion. Establishing this rigorously is highly nontrivial due to the degenerate structure of the Navier-Stokes equations. To overcome this difficulty, we first derive a pointwise estimate for the density and then obtain spatial decay estimates for the velocity gradient via carefully constructed weighted high-order energy estimates and interpolation inequalities. The resulting decay rate is sufficiently rapid to compensate for the singular growth of the density, leading to uniform-in-time control of the viscous terms and ultimately to the formation of implosion.
Paper Structure (40 sections, 36 theorems, 490 equations, 1 figure)

This paper contains 40 sections, 36 theorems, 490 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations, Reformulations, and Outline of the Proof
  3. Notations
  4. Notations on coordinates and operators
  5. Notations on function spaces
  6. Other notations
  7. Reformulation
  8. Reformulation based on the self-similar scaling
  9. Reformulation based on $C^\infty$ smooth spherically symmetric profiles of the steady Euler equations
  10. Outline of the Proof
  11. Spatial decay estimates
  12. Lower-order temporal decay estimates
  13. $K$-th order weighted estimates
  14. Control of the unstable modes of $(\overset{\approx}{Q},\overset{\approx}{U})$ by the selection of initial data.
  15. Local-In-Time Well-Posedness of the Three-Dimensional Smooth Solutions
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $1 <\gamma <1+\frac{2}{\sqrt{3}}$, let $\delta^*(\gamma)$ be a constant given by and let $(\overline{Q}, \overline{U})$ be the $C^\infty$ self-similar profile solving Profile obtained in Lemma lem:existofselfsimilar (see Appendix appendix B) when the scaling parameter $\Lambda$ introduced in scaling1--scaling-coordinat1 satisfies range of Lambda--def:Lambda*. Then, for two const there are $C^

Figures (1)

  • Figure 2: Phase portrait trajectory from $P_\infty$ to $P_0$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2: Choice and Hierarchy of Parameters
  • Theorem 3.1
  • Theorem 3.2
  • ...and 49 more