[Paper Review] Nonlinearly Exponential Stability of Compressible Navier-Stokes System with Degenerate Heat-Conductivity

Bin Huang, Xiaoding Shi|arXiv (Cornell University)|Sep 3, 2018

Navier-Stokes equation solutions3 citations

TL;DR

This paper establishes the nonlinear exponential stability of strong solutions to the one-dimensional compressible Navier-Stokes system with degenerate heat conductivity, where thermal conductivity is proportional to a positive power of temperature and viscosity is constant. By deriving uniform a priori estimates and leveraging energy methods, the authors prove that the specific volume and temperature remain uniformly bounded away from zero and infinity, and the solution converges exponentially to the equilibrium state as time tends to infinity, extending Kazhikhov's results to the degenerate nonlinear case with arbitrary large initial data.

ABSTRACT

We study the large-time behavior of strong solutions to the one-dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. Both the specific volume and the temperature are proved to be bounded from below and above independently of time. Moreover, it is shown that the global solution is nonlinearly exponentially stable as time tends to infinity. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977); Kazhikhov. Boundary Value Problems for Hydrodynamical Equations, 50 (1981)] and can be arbitrarily large. Therefore, our result can be regarded as a natural generalization of the Kazhikhov's ones for the constant heat conductivity case to the degenerate and nonlinear one.

Motivation & Objective

• To investigate the large-time behavior of strong solutions to the one-dimensional compressible Navier-Stokes system with degenerate heat conductivity.
• To extend Kazhikhov’s nonlinear exponential stability result from the constant heat conductivity case to the physically relevant case where heat conductivity depends nonlinearly on temperature.
• To establish global existence and stability of strong solutions under minimal assumptions on initial data, including arbitrarily large initial perturbations.
• To prove uniform boundedness of specific volume and temperature independently of time, ensuring the solution remains in a non-degenerate regime.

Proposed method

• Formulate the compressible Navier-Stokes system in Lagrangian coordinates with viscosity constant and heat conductivity proportional to a positive power of temperature.
• Apply energy estimates and Sobolev embedding to derive uniform bounds on the solution in $ H^1 $-norm for all time.
• Use the maximum principle and pointwise estimates to control the lower and upper bounds of specific volume and temperature.
• Establish decay estimates for the velocity and temperature gradients via integration over time intervals and use of $ L^2 $-type energy identities.
• Leverage the decay of $ H^1 $-norm of spatial derivatives to prove exponential convergence to the equilibrium state.
• Utilize the fact that the $ L^2 $-norm of the velocity decays to zero as time goes to infinity, implying nonlinear exponential stability.

Experimental results

Research questions

• RQ1 Can the nonlinear exponential stability of the compressible Navier-Stokes system be extended to the case of degenerate heat conductivity proportional to a positive power of temperature?
• RQ2 Does the global strong solution remain uniformly bounded in time for initial data with arbitrary size, even when heat conductivity degenerates?
• RQ3 Can the solution remain bounded away from zero in both specific volume and temperature for all time, despite the degeneracy in heat conductivity?
• RQ4 Is the convergence of the solution to the equilibrium state exponential in time under these conditions?
• RQ5 Can the assumptions on initial data be relaxed compared to previous works, particularly in the degenerate heat conductivity regime?

Key findings

• The specific volume $ v $ and temperature $ heta $ are uniformly bounded from below and above independently of time, with bounds depending only on initial data and physical parameters.
• The global strong solution exists for all time and satisfies $ v, heta o ext{constant} $ exponentially fast as $ t o ty $, with $ C^{-1} \ leq v(x,t) \ leq C $ and $ C^{-1} \ leq heta(x,t) \ leq C $ for all $ (x,t) \in (0,1) \times (0,\infty) $.
• The solution $ (v,u,\theta) $ converges to the equilibrium state $ (1,0,1) $ exponentially in the $ H^1 $-norm: $ \|(v-1,u,\theta-1)(\cdot,t)\|_{H^1} \leq C e^{-\eta_0 t} $ for some $ \eta_0 > 0 $.
• The $ L^2 $-norm of the velocity $ u $ decays to zero as $ t \to \infty $, which is essential for proving exponential stability.
• The $ L^2 $-norms of $ u_x $, $ \theta_x $, $ u_t $, $ \theta_t $, $ v_{xt} $, $ u_{xx} $, and $ \theta_{xx} $ are all uniformly bounded in time.
• The decay of the $ H^1 $-norm of the solution gradients implies that the solution approaches the equilibrium state at an exponential rate, confirming nonlinear exponential stability.