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Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations

Yongteng Gu, Xiangdi Huang

TL;DR

The paper proves global classical solutions for the multi-dimensional non-isentropic compressible Navier–Stokes equations with transported entropy under radial symmetry, extending the BD entropy framework beyond isentropic cases. It develops a BD-type entropy inequality in Lagrangian coordinates with the effective velocity $w$, and derives a hierarchy of a priori estimates (from $L^2$ to $L^ abla$∞ and higher-order Sobolev bounds) to control density, velocity, and entropy. The main results show global existence for $N=2$ with $eta>1$ (i.e., $ abla ho$ controlled) and for $N=3$, $1<oldsymbol{ ho}oldsymbol{ ext{gamma}}<3$, with density bounded away from vacuum and solvability in the transported-entropy setting. A corollary addresses the constant-entropy (shallow water) case, yielding uniform density bounds and large-time behavior. Overall, the work broadens the BD entropy methodology to non-isentropic fluids and higher dimensions, enabling global classical solutions for large initial data in spherically symmetric domains.

Abstract

In 1871, Saint-Venant introduced the shallow water equations. Since then, the global classical solutions for arbitrary large initial data of the multi-dimensional viscous Saint-Venant system have remained a well-known open problem. It was only recently that [Huang-Meng-Zhang, http:arXiv:2512.15029, 2025], under the assumption of radial symmetry, first proved the existence of global classical solutions for arbitrary large initial data to the initial-boundary value problem of the two-dimensional viscous shallow water equations. At the same time, [Chen-Zhang-Zhu, http:arXiv:2512.18545, 2025] also independently proved the existence of global large solutions to the Cauchy problem of this system. Notably, in the work of Huang-Meng-Zhang, they also established the existence of global classical solutions for arbitrary large initial data to the isentropic compressible Navier-Stokes equations satisfying the BD entropy equality in both two and three dimensions, and the viscous shallow water equations are precisely a specific class of isentropic compressible fluids subject to the BD entropy equality. In this paper, we prove a new BD entropy inequality for a class of non-isentropic compressible fluids, which can be regarded as a generalization of the shallow water equations with transported entropy. Employing new estimates on the lower bound of density different from that of Huang-Meng-Zhang's work, we show the "viscous shallow water system with transport entropy" will admit global classical solutions for arbitrary large initial data to the spherically symmetric initial-boundary value problem in both two and three dimensions. Our results also relax the restrictions on the dimension and adiabatic index imposed in Huang-Meng-Zhang's work on the shallow water equations, extending the range from $N=2,\ γ\ge \frac{3}{2}$ to $N=2,\ γ> 1$ and $N=3,\ 1<γ<3$.

Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations

TL;DR

The paper proves global classical solutions for the multi-dimensional non-isentropic compressible Navier–Stokes equations with transported entropy under radial symmetry, extending the BD entropy framework beyond isentropic cases. It develops a BD-type entropy inequality in Lagrangian coordinates with the effective velocity , and derives a hierarchy of a priori estimates (from to ∞ and higher-order Sobolev bounds) to control density, velocity, and entropy. The main results show global existence for with (i.e., controlled) and for , , with density bounded away from vacuum and solvability in the transported-entropy setting. A corollary addresses the constant-entropy (shallow water) case, yielding uniform density bounds and large-time behavior. Overall, the work broadens the BD entropy methodology to non-isentropic fluids and higher dimensions, enabling global classical solutions for large initial data in spherically symmetric domains.

Abstract

In 1871, Saint-Venant introduced the shallow water equations. Since then, the global classical solutions for arbitrary large initial data of the multi-dimensional viscous Saint-Venant system have remained a well-known open problem. It was only recently that [Huang-Meng-Zhang, http:arXiv:2512.15029, 2025], under the assumption of radial symmetry, first proved the existence of global classical solutions for arbitrary large initial data to the initial-boundary value problem of the two-dimensional viscous shallow water equations. At the same time, [Chen-Zhang-Zhu, http:arXiv:2512.18545, 2025] also independently proved the existence of global large solutions to the Cauchy problem of this system. Notably, in the work of Huang-Meng-Zhang, they also established the existence of global classical solutions for arbitrary large initial data to the isentropic compressible Navier-Stokes equations satisfying the BD entropy equality in both two and three dimensions, and the viscous shallow water equations are precisely a specific class of isentropic compressible fluids subject to the BD entropy equality. In this paper, we prove a new BD entropy inequality for a class of non-isentropic compressible fluids, which can be regarded as a generalization of the shallow water equations with transported entropy. Employing new estimates on the lower bound of density different from that of Huang-Meng-Zhang's work, we show the "viscous shallow water system with transport entropy" will admit global classical solutions for arbitrary large initial data to the spherically symmetric initial-boundary value problem in both two and three dimensions. Our results also relax the restrictions on the dimension and adiabatic index imposed in Huang-Meng-Zhang's work on the shallow water equations, extending the range from to and .
Paper Structure (17 sections, 20 theorems, 189 equations)

This paper contains 17 sections, 20 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. A priori estimates: Lower-Order Estimates for $N=2$
  4. $L^2$ estimates for velocity and effective velocity
  5. $L^4$ estimates for velocity and effective velocity
  6. Upper bound estimates for the density
  7. $L^{\infty}$ estimates for velocity and effective velocity
  8. Lower bound estimates for the density
  9. A priori estimates: Lower-Order Estimates for $N=3$
  10. $L^2$ estimates for velocity and effective velocity
  11. Upper Bound Estimates for the Density
  12. $L^{\infty}$ estimates for velocity and effective velocity
  13. Lower Bound Estimate for the Density
  14. A priori estimates: Higher-Order estimates
  15. Proof of the Main Results
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that and the radially symmetric initial data $(\rho_0, \mathbf{u_0},s_0)$ satisfies Then, there exists a unique global radially symmetric classical solution to the initial-boundary value problem 1.1,1.2 and 1.3 satisfying, for any $(x,t) \in \Omega \times[0,T]$, where the constant $C(T) > 0$ depends on the initial data and $T$.

Theorems & Definitions (38)

  • Definition 1.1: Global classical solution
  • Theorem 1.1: Global classical solutions for $N=2$
  • Theorem 1.2: Global classical solutions for $N=3$
  • Remark 1.1
  • Corollary 1.1: Uniform bounds on density and large-time behavior
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 28 more