[Paper Review] Global Strong Solutions for 3D Viscous Incompressible Heat Conducting Navier-Stokes Flows with Non-negative Density

Xin Zhong|arXiv (Cornell University)|May 17, 2016

Advanced Mathematical Physics Problems16 references3 citations

TL;DR

This paper establishes the existence of global strong solutions for three-dimensional viscous, incompressible, heat-conducting Navier-Stokes flows with non-negative density, including vacuum. By applying delicate energy estimates and leveraging regularity properties of the Stokes system and elliptic equations, it proves global existence under Serrin-type conditions on velocity and smallness assumptions on initial data.

ABSTRACT

We are concerned with an initial boundary value problem for the nonhomogeneous heat conducting Navier-Stokes flows with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier-Stokes flows. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation.

Motivation & Objective

• To address the existence of global strong solutions for 3D nonhomogeneous heat-conducting Navier-Stokes equations with non-negative density.
• To analyze the case where the initial density may vanish (i.e., vacuum is allowed).
• To establish global existence under Serrin-type integrability conditions on the velocity field.
• To prove uniqueness and global existence under smallness conditions on initial data.

Proposed method

• Employing delicate energy estimates to control the nonlinear terms in the Navier-Stokes system.
• Utilizing regularity properties of the Stokes system to handle the velocity and pressure components.
• Applying elliptic regularity theory to the temperature and pressure equations.
• Using a priori estimates to bound the solution norms uniformly in time.
• Establishing existence via a Galerkin approximation and limit passage under smallness assumptions.
• Combining Serrin-type conditions with small initial data to ensure global existence.

Experimental results

Research questions

• RQ1 Under what conditions does a global strong solution exist for 3D nonhomogeneous heat-conducting Navier-Stokes flows with vacuum?
• RQ2 Can the Serrin condition on velocity guarantee global existence when density is non-negative and possibly zero?
• RQ3 What smallness conditions on initial data ensure the existence of a unique global strong solution?

Key findings

• Global strong solutions exist for 3D viscous incompressible heat-conducting Navier-Stokes flows with non-negative density if the velocity satisfies the Serrin condition.
• Under suitable smallness assumptions on initial data, a unique global strong solution exists.
• The method relies on energy estimates and regularity results for the Stokes system and elliptic equations.
• The analysis holds even when the initial density vanishes in some regions (i.e., vacuum is allowed).
• The solution remains strong and globally defined under the stated conditions, ensuring regularity and uniqueness.