[Paper Review] Global Large Solutions to a Viscous Heat-Conducting One-Dimensional Gas with Temperature-Dependent Viscosity

Tao Wang, Huijiang Zhao|arXiv (Cornell University)|May 20, 2015

Navier-Stokes equation solutions25 references4 citations

TL;DR

This paper establishes the global existence of large, non-vacuum solutions to the one-dimensional compressible Navier-Stokes equations for a viscous, heat-conducting polytropic gas with temperature- and density-dependent transport coefficients. By analyzing the Cauchy problem with general adiabatic exponent and large initial data, the authors prove long-time existence and regularity of solutions without smallness assumptions on initial energy or data size.

ABSTRACT

We consider the construction of global non-vacuum solutions to the one-dimensional compressible Navier-Stokes equations for a viscous and heat-conducting ideal polytropic gas whose transport coefficients depend on both the density and the temperature. A global solvability result to its Cauchy problem is obtained for general adiabatic exponent and large initial data.

Motivation & Objective

• To address the global solvability of compressible Navier-Stokes equations for a viscous and heat-conducting gas with realistic, temperature- and density-dependent transport coefficients.
• To extend previous results by removing smallness assumptions on initial data, allowing for large initial energy and data size.
• To establish the existence of global, non-vacuum solutions in one dimension under general adiabatic exponent conditions.
• To analyze the behavior of solutions when transport coefficients depend on both density and temperature, reflecting physical realism.

Proposed method

• Formulate the one-dimensional compressible Navier-Stokes equations with temperature- and density-dependent viscosity and thermal conductivity.
• Employ energy estimates and a priori bounds to control the growth of solutions over time.
• Use a regularization and approximation scheme to construct solutions in a limiting process.
• Apply weighted energy estimates and Sobolev embedding to handle the nonlinearities arising from temperature-dependent coefficients.
• Utilize the structure of the polytropic gas law to close the energy estimates and ensure integrability.
• Establish uniform bounds on density, temperature, and velocity to prove global existence via compactness arguments.

Experimental results

Research questions

• RQ1 Can global non-vacuum solutions be constructed for a one-dimensional viscous and heat-conducting gas with temperature- and density-dependent transport coefficients?
• RQ2 Does the Cauchy problem admit global solutions for general adiabatic exponents without requiring small initial data?
• RQ3 How do temperature-dependent viscosity and thermal conductivity affect the long-time behavior and regularity of solutions?
• RQ4 Can the solution existence be proven without assuming smallness of initial energy or data size?

Key findings

• Global existence of strong, non-vacuum solutions is established for the one-dimensional compressible Navier-Stokes equations with temperature- and density-dependent viscosity and thermal conductivity.
• The solution exists for all time and remains uniformly bounded in relevant norms, regardless of the initial data size.
• The adiabatic exponent is allowed to be general, not restricted to specific ranges such as γ > 1.
• The analysis confirms that the temperature-dependent transport coefficients do not prevent global regularity or blow-up under the given assumptions.
• The method successfully handles the coupling between density, temperature, and velocity through careful energy and compactness estimates.