[Paper Review] Global solution and time decay of the compressible Euler-Maxwell system in R3

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

Navier-Stokes equation solutions18 references3 citations

TL;DR

This paper establishes the global existence of unique solutions to the compressible Euler-Maxwell system in $[\mathbb{R}^3$ using only small $H^3$ norm of initial data, without requiring smallness of higher-order derivatives. By applying a regularity interpolation technique, it derives various decay rates for the solution and its derivatives when initial data lies in $\Dot{H}^{-s}$ or $\Dot{B}_{2,\infty}^{-s}$ spaces, yielding $L^p$--$L^2$ decay estimates without assuming small $L^p$ norms.

ABSTRACT

We study the global existence and large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in $\mathhbb{R}^3$. The previous works mainly assumed that the $H^N$ $(N\ge3)$ and $L^1$ norms of the initial data are sufficiently small. In this paper, we first construct the global unique solution by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to $\Dot{H}^{-s}$ ($0\le s<3/2$) or $\dot{B}_{2,\infty}^{-s}$ ($0< s\le3/2$), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the $L^p$--$L^2$ $(1\le p\le 2)$ type of the decay rates follow without requiring that the $L^p$ norm of initial data is small.

Motivation & Objective

• To establish global existence of solutions to the compressible Euler-Maxwell system in $\mathbb{R}^3$ under weaker initial data assumptions than previous works.
• To analyze the large-time behavior of solutions near a constant equilibrium, particularly decay rates of the solution and its derivatives.
• To derive $L^p$--$L^2$ type decay estimates without requiring the $L^p$ norm of initial data to be small.
• To extend decay rate results to initial data in negative-order Sobolev or Besov spaces, including $\dot{H}^{-s}$ and $\dot{B}_{2,\infty}^{-s}$ for $0 \leq s < 3/2$.

Proposed method

• Constructing global unique solutions using only the smallness of the $H^3$ norm of initial data, even when higher-order derivatives are large.
• Applying a regularity interpolation technique to bridge the gap between $H^3$ regularity and decay estimates in negative-order function spaces.
• Employing energy estimates and spectral analysis to control the evolution of the solution and its derivatives over time.
• Utilizing the structure of the Euler-Maxwell system to derive decay properties through the interplay between fluid dynamics and electromagnetic fields.
• Deriving decay rates in $L^p$--$L^2$ type by exploiting the regularity of initial data in negative-order spaces.
• Establishing that $L^p$--$L^2$ decay holds without requiring smallness of the $L^p$ norm of initial data, relying instead on the regularity of the data in negative-order spaces.

Experimental results

Research questions

• RQ1 Can global solutions be constructed for the compressible Euler-Maxwell system in $\mathbb{R}^3$ with only $H^3$-small initial data, without assuming smallness of higher-order derivatives?
• RQ2 What decay rates can be obtained for the solution and its higher-order derivatives when initial data belongs to negative-order Sobolev spaces $\dot{H}^{-s}$ or Besov spaces $\dot{B}_{2,\infty}^{-s}$?
• RQ3 Can $L^p$--$L^2$ type decay estimates be derived without assuming the $L^p$ norm of initial data is small?
• RQ4 How does the regularity interpolation technique enable the derivation of refined decay rates from $H^3$-initial data?
• RQ5 What is the role of the negative-order norm structure in obtaining optimal decay behavior for the solution and its derivatives?

Key findings

• Global unique solutions exist for the compressible Euler-Maxwell system in $\mathbb{R}^3$ under the sole assumption that the $H^3$ norm of initial data is small.
• Decay rates for the solution and its higher-order derivatives are derived when initial data lies in $\dot{H}^{-s}$ for $0 \leq s < 3/2$ or $\dot{B}_{2,\infty}^{-s}$ for $0 < s \leq 3/2$.
• The $L^p$--$L^2$ type decay estimates are obtained without requiring the $L^p$ norm of initial data to be small, relying instead on the regularity in negative-order spaces.
• The decay rates depend on the negative-order regularity of the initial data, with faster decay for larger $s$ in the negative-order spaces.
• The regularity interpolation technique successfully connects $H^3$-initial data to optimal decay behavior in $L^p$ and higher-order norms.
• The results extend previous works by relaxing the smallness assumptions on higher-order derivatives and $L^p$ norms of initial data.