[Paper Review] Decay of dissipative equations and negative Sobolev spaces

Yan Guo, Yanjin Wang|arXiv (Cornell University)|Nov 24, 2011

Computational Fluid Dynamics and Aerodynamics23 references4 citations

TL;DR

This paper develops a novel energy method to establish optimal time decay rates for solutions to dissipative equations such as the heat equation, compressible Navier-Stokes, and Boltzmann equations. By combining scaled energy estimates with Sobolev interpolation involving negative Sobolev norms, it proves that the decay rates of higher-order spatial derivatives match those of the linearized system, with decay rates quantified by powers of $(1+t)^{-(\ell+s)/2}$, where $s$ measures the initial data's integrability in negative Sobolev space.

ABSTRACT

We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

Motivation & Objective

• To establish optimal time decay rates for solutions to dissipative equations like Navier-Stokes and Boltzmann equations.
• To overcome limitations of prior pure energy methods that failed to achieve optimal decay rates.
• To replace the $L^p$ norm assumption in previous works with the more robust negative Sobolev norm $\dot{H}^{-s}$.
• To unify the analysis of decay rates across different dissipative systems using a single framework based on energy estimates and interpolation.
• To demonstrate that negative Sobolev norms are preserved under time evolution and enhance decay estimates.

Proposed method

• Introduce a family of scaled energy estimates with minimal derivative counts to track decay behavior across different regularity levels.
• Use Sobolev interpolation between negative Sobolev norms $\dot{H}^{-s}$ and positive Sobolev norms $H^N$ to bridge regularity gaps.
• Apply the interpolation inequality $\|\nabla^\ell f\|_{L^2} \lesssim \|\nabla^{\ell+1}f\|_{L^2}^{1-\theta} \|\Lambda^{-s}f\|_{L^2}^\theta$ with $\theta = 1/(\ell+1+s)$ to relate high-order derivatives to low-regularity data.
• Establish a differential inequality of the form $\frac{d}{dt}\|\nabla^\ell u\|_{L^2}^2 + C_0 \|\nabla^\ell u\|_{L^2}^{2(1+1/(\ell+s))} \leq 0$ to derive time decay.
• Use the fact that $\|\Lambda^{-s}u(t)\|_{L^2}$ is non-increasing in time to preserve the negative norm and strengthen decay estimates.
• Apply the method to classical systems: heat equation, compressible Navier-Stokes, and Boltzmann equation, showing consistent decay behavior.

Experimental results

Research questions

• RQ1 Can optimal time decay rates for higher-order spatial derivatives of solutions to dissipative equations be proven via a pure energy method without spectral analysis or $L^p$ assumptions?
• RQ2 How can negative Sobolev norms $\dot{H}^{-s}$ be used to replace $L^p$ integrability assumptions in decay analysis?
• RQ3 What is the role of Sobolev interpolation between negative and positive Sobolev norms in deriving sharp decay estimates?
• RQ4 Can the decay rate of $\|\nabla^\ell u(t)\|_{L^2}$ be shown to be $\lesssim (1+t)^{-(\ell+s)/2}$ for $\ell \in [-s, N]$?
• RQ5 Is the negative Sobolev norm $\|\Lambda^{-s}u(t)\|_{L^2}$ preserved or non-increasing along time evolution in these systems?

Key findings

• The optimal decay rate for $\|\nabla^\ell u(t)\|_{L^2}$ is $\lesssim (1+t)^{-(\ell+s)/2}$ for $\ell \in [-s, N]$, where $s \geq 0$ quantifies the initial data's regularity in $\dot{H}^{-s}$.
• The negative Sobolev norm $\|\Lambda^{-s}u(t)\|_{L^2}$ is non-increasing in time, which is crucial for enhancing decay estimates.
• The method achieves optimal decay rates without relying on linear decay analysis or $L^p$ norm preservation, overcoming limitations of prior energy methods.
• For the heat equation, the decay rate $\|\nabla^\ell u(t)\|_{L^2} \leq C_0 (1+t)^{-(\ell+s)/2}$ is proven under initial data $u_0 \in H^N \cap \dot{H}^{-s}$.
• The same framework applies to the compressible Navier-Stokes and Boltzmann equations, yielding the same optimal decay rates as the linearized systems.
• The use of interpolation between $\dot{H}^{-s}$ and $H^N$ allows control of intermediate regularity levels without requiring $L^p$ integrability of initial data.