[Paper Review] Decay estimates of solutions to the compressible Euler-Maxwell system in R3

Zhong Tan, Yanjin Wang+1 more|arXiv (Cornell University)|Jul 10, 2012

Navier-Stokes equation solutions21 references3 citations

TL;DR

This paper establishes sharp time decay rates for solutions to the compressible Euler-Maxwell system in $ℝ^3$ near a constant equilibrium, using energy estimates and regularity interpolation. It proves that solutions and their spatial derivatives decay at rates $(1+t)^{-(k+3+s)/2}$ for $k$-th order derivatives, even when initial data have large higher-order norms, under small $H^3$ norm and $̂dot{H}^{-s}$ or $̂dot{B}_{2,∞}^{-s}$ regularity with $0 \leq s < 3/2$. The results hold without requiring small $L^p$ norms of initial data.

ABSTRACT

We study the large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in $ 3$. We first refine a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If the initial data belongs to $\Dot{H}^{-s}$ ($0\le s&lt;3/2$) or $\dot{B}_{2,\infty}^{-s}$ ($0

Motivation & Objective

• To analyze the large-time behavior of solutions to the compressible Euler-Maxwell system in $ℝ^3$ near a constant equilibrium state.
• To derive decay estimates for the solution and its higher-order spatial derivatives under minimal smallness assumptions on the initial data.
• To establish decay rates without requiring smallness of the $L^p$ norm of initial data, relying instead on regularity in negative-order Sobolev or Besov spaces.
• To refine global existence results by allowing large higher-order derivatives in initial data, provided the $H^3$ norm is small.

Proposed method

• Employing a regularity interpolation technique to bridge low and high-order Sobolev norms in the energy method framework.
• Applying energy estimates to a reformulated system with rescaled variables, reducing the problem to a symmetric hyperbolic system with relaxation.
• Deriving a priori estimates for the $L^2$-norms of $k$-th order spatial derivatives of the density $n$ and a derived potential $ψ$, using commutator estimates and Sobolev embeddings.
• Using the Gronwall inequality on energy-dissipation estimates to obtain time-decay bounds, iteratively improving decay rates by assuming higher initial regularity.
• Estimating nonlinear terms via product rules and commutator estimates, controlling them using smallness of $H^3$ norm and $L^∞$ norms of lower-order derivatives.
• Establishing equivalence between the energy functional and the $H^k$-norm of the solution, enabling decay rate transfer to the original variables.

Experimental results

Research questions

• RQ1 What decay rates can be established for solutions to the compressible Euler-Maxwell system in $ℝ^3$ when initial data have only small $H^3$ norm but possibly large higher-order derivatives?
• RQ2 Can decay estimates be derived without assuming smallness of the $L^p$ norm of initial data, especially for $1 \leq p \leq 2$?
• RQ3 How do decay rates depend on the regularity of initial data in negative-order Sobolev or Besov spaces, such as $\dot{H}^{-s}$ or $\dot{B}_{2,\infty}^{-s}$ with $0 \leq s < 3/2$?
• RQ4 What is the role of the relaxation term in enabling global existence and decay under minimal smallness assumptions?
• RQ5 Can iterative refinement of decay estimates be achieved by assuming higher initial regularity, leading to improved decay rates?

Key findings

• Solutions to the compressible Euler-Maxwell system in $ℝ^3$ decay at the rate $(1+t)^{-(k+3+s)/2}$ for the $L^2$-norm of the $k$-th order spatial derivatives of the density $n$ and the potential $\psi$, under small $H^3$ norm and $\dot{H}^{-s}$ or $\dot{B}_{2,\infty}^{-s}$ regularity with $0 \leq s < 3/2$.
• The decay rates are uniform and do not require the $L^p$-norm of the initial data to be small, which extends previous results that relied on such smallness.
• By assuming $N \geq 2k+8+s$ initial regularity, the paper derives the decay estimate $\|\nabla^k(n,\psi)(t)\|_{L^2} \lesssim C_0 (1+t)^{-(k+3+s)/2}$, which is sharp under the given assumptions.
• Iterative refinement of the decay estimate is possible: assuming $N \geq 2k+12+s$, the decay improves to $\|\nabla^k(n,\psi)(t)\|_{L^2} \lesssim C_0 (1+t)^{-(k/2 + 7/4 + s)}$, demonstrating improved decay through higher regularity.
• The energy method combined with regularity interpolation allows control of nonlinear terms via smallness of $H^3$ norm and $L^\infty$ norms of lower-order derivatives, even when higher-order derivatives are large.
• The results confirm that the relaxation term is essential for obtaining global existence and decay under minimal smallness conditions, as the non-relaxation case remains significantly more challenging.