[Paper Review] Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum

Shijin Ding, Huanyao Wen+2 more|arXiv (Cornell University)|Mar 8, 2011

Navier-Stokes equation solutions12 references3 citations

TL;DR

This paper establishes the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations in bounded or exterior domains of $[\mathbb{R}^n$ for $n \geq 2$, even when initial data are large and vacuum is present. The authors develop novel energy estimates and mathematical techniques to achieve higher regularity than previous works, marking the first result on global classical solutions with large initial data and vacuum in higher dimensions.

ABSTRACT

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations with vacuum in a bounded domain or exterior domain Ω of Rn(n >= 2). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1-28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91-129; S.J. Ding, H.Y.Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696-1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension

Motivation & Objective

• To establish the existence and uniqueness of global classical solutions for compressible isentropic Navier-Stokes equations with large initial data in spherically symmetric settings.
• To address the challenge of vacuum presence in higher-dimensional domains ($n \geq 2$), where previous results were limited to small data or one dimension.
• To improve the regularity of solutions beyond prior works, particularly in the presence of vacuum and large initial data.
• To extend previous results on local or strong solutions to global classical solutions in higher dimensions with minimal assumptions on initial data.

Proposed method

• Derive new energy estimates tailored for spherically symmetric solutions with vacuum, using weighted $L^2$ norms involving $r^m$ with $m = n-1$.
• Employ a priori estimates up to fifth-order spatial derivatives of density and velocity, leveraging the structure of the compressible Navier-Stokes system in radial form.
• Apply Gronwall’s inequality to control time-dependent energy norms and close the a priori estimates for global existence.
• Use a regularization procedure with $δ$-dependent solutions and take limits as $\delta \to 0$ to handle vacuum and ensure regularity.
• Establish uniform bounds on $\rho$, $\rho^\gamma$, $u$, and their derivatives in Sobolev and $L^\infty$ spaces over time.
• Utilize the Lamé operator and compatibility conditions to ensure the existence of initial data satisfying the necessary regularity and boundary constraints.

Experimental results

Research questions

• RQ1 Can global classical solutions exist for the compressible Navier-Stokes equations with large initial data and vacuum in higher dimensions ($n \geq 2$)?
• RQ2 What new mathematical techniques are required to control the loss of regularity near vacuum in spherically symmetric flows?
• RQ3 How can higher-order energy estimates be constructed to ensure global existence despite vacuum formation and large initial data?
• RQ4 Is it possible to achieve better regularity than previous works (e.g., $H^5$ for density and $H^6$ for velocity) under the same conditions?
• RQ5 Can the solution framework be extended from annular domains to balls or exterior domains with vacuum?

Key findings

• The paper proves the existence and uniqueness of global spherically symmetric classical solutions to the compressible Navier-Stokes equations in bounded or exterior domains for $n \geq 2$, even with large initial data and vacuum.
• The solutions satisfy $\|\rho\|_{H^5}, \|\rho^\gamma\|_{H^5} \leq c$, $\|u\|_{H^5} \cap L^2(H^6) \leq c$, and $\|u_t\|_{H^1} \cap L^2(H^3) \leq c$, indicating improved regularity compared to prior works.
• The authors establish uniform bounds on $\int_I r^m |\partial_r^5 \rho|^2 + |\partial_r^5 (\rho^\gamma)|^2 \leq c$, confirming fifth-order regularity of density and pressure.
• The solution satisfies $\rho \geq \delta / c > 0$ in the regularized setting, and the limit as $\delta \to 0$ yields a solution with vacuum allowed.
• The method prevents the solution from achieving $L^\infty([0,T]; D^6(\Omega))$ or $L^2([0,T]; D^8(\Omega))$ regularity due to vacuum, even with smooth initial data.
• The results represent the first global classical solution with large initial data and vacuum in higher dimensions, extending prior results limited to 1D or small data.