[Paper Review] Global classical solution to the Cauchy problem of 2D baratropic compressible Navier-Stokes system with large initial data

Jingchi Huang, Chao Wang|arXiv (Cornell University)|Jul 20, 2014

Navier-Stokes equation solutions21 references3 citations

TL;DR

This paper establishes the global existence and uniqueness of classical solutions to the 2D barotropic compressible Navier-Stokes equations with large initial data, under the condition that the bulk viscosity is $λ = \rho^\beta$ with $\beta > 1$. By deriving a priori bounds on density and velocity gradients through energy estimates and logarithmic Sobolev inequalities, the authors extend local solutions globally in time without requiring smallness of initial data in $L^\infty$ or $L^2$ norms.

ABSTRACT

For periodic initial data with initial density, we establish the global existence and uniqueness of strong and classical solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\lam=ρ^{\b}$ with $\b>1.$

Motivation & Objective

• To establish global existence and uniqueness of classical solutions for the 2D compressible Navier-Stokes equations with large initial data.
• To remove the restriction $\beta > 3$ or $\beta > 4/3$ previously required in the literature for global well-posedness.
• To allow large initial density oscillations by proving uniform upper bounds on the density and its gradients.
• To extend local strong solutions to global solutions using a contradiction argument based on a priori estimates.
• To prove the result under minimal assumptions: $\beta > 1$, $\gamma > 1$, initial density in $W^{1,q}$, and velocity in $H^2$.

Proposed method

• Derive a priori bounds on the density using a transport equation for $\Phi = (2\mu + \lambda(\rho))\nabla\rho$ and integrate against $|\Phi|^{q-2}\Phi$.
• Employ logarithmic Sobolev inequalities to control $\|\nabla u\|_{L^\infty}$ in terms of $\|\nabla^2 u\|_{L^q}$ and $\|\nabla u\|_{L^2}$.
• Use the material derivative $\dot{u} = u_t + u \cdot \nabla u$ to derive energy estimates for $\|\rho^{1/2} \dot{u}\|_{L^2}$ and $\|\nabla \dot{u}\|_{L^2}$.
• Apply Gronwall's inequality to the density gradient estimate $\|\nabla \rho\|_{L^q}$ after bounding $\|\rho \dot{u}\|_{L^q}$ via $L^2(0,T;L^q)$ control.
• Establish higher-order estimates for $\|\nabla^2 u\|_{L^q}$ using the relation $\|\nabla^2 u\|_{L^q} \lesssim \|\nabla \text{div} u\|_{L^q} + \|\nabla \omega\|_{L^q}$ and bounds on $\|\nabla \rho\|_{L^q}$.
• Use a contradiction argument: assume finite lifespan $T^*$, then show that uniform bounds on $\|\rho\|_{L^\infty}$ and $\|\nabla \rho\|_{L^q}$ allow extension beyond $T^*$, contradicting maximality.

Experimental results

Research questions

• RQ1 Can global classical solutions exist for the 2D compressible Navier-Stokes equations with large initial data when the bulk viscosity is $\lambda = \rho^\beta$?
• RQ2 What is the minimal requirement on $\beta$ for global well-posedness without smallness assumptions on initial data?
• RQ3 Can uniform upper bounds on the density be established for large initial data when $\beta > 1$?
• RQ4 How can the blow-up criterion be controlled to extend local solutions globally?
• RQ5 Can the $L^\infty$-norm of the density be controlled using only $L^q$-bounds on $\nabla \rho$ and $L^2(0,T;L^q)$-bounds on $\rho \dot{u}$?

Key findings

• The paper proves the global existence and uniqueness of strong and classical solutions for the 2D compressible Navier-Stokes equations with large initial data.
• The result holds under the condition $\beta > 1$ for the bulk viscosity $\lambda = \rho^\beta$, improving upon previous results that required $\beta > 3$ or $\beta > 4/3$.
• A uniform upper bound on the density is established via a transport equation for $\Phi = (2\mu + \lambda(\rho))\nabla\rho$, which is critical for extending solutions globally.
• The $L^q$-norm of $\nabla \rho$ is bounded uniformly in time using Gronwall's inequality after estimating $\|\rho \dot{u}\|_{L^q}$ and $\|\nabla u\|_{L^\infty}$ via logarithmic Sobolev embeddings.
• Higher-order estimates for $\|\nabla^2 u\|_{L^q}$ are derived from bounds on $\|\nabla \text{div} u\|_{L^q}$ and $\|\nabla \omega\|_{L^q}$, which are controlled by $\|\nabla \rho\|_{L^q}$ and $\|\rho \dot{u}\|_{L^q}$.
• The contradiction argument confirms that the solution can be extended beyond any finite time $T^*$, proving global existence under the stated conditions.