Incompressible limit of nonisentropic Hookean elastodynamics

Abstract

We study the incompressible limit of the compressible nonisentropic Hookean elastodynamics with general initial data in the whole space [Formula: see text]. First, we obtain the uniform estimates of the solutions in [Formula: see text] for s > d/2 + 1 being even and the existence of classic solutions on a time interval independent of the Mach number. Then, we prove that the solutions converge to the incompressible elastodynamic equations as the Mach number tends to zero.

Low Mach number limit of nonisentropic inviscid Hookean elastodynamics

“…Recently, Wang 19 also studied the incompressible limit of nonisentropic Hookean elastodynamics for ill‐prepared initial data in the whole space $$$ {\mathbb{R}}^d $$$. One of the key assumptions in Wang 19 is that the index $$$ s\ge 4 $$$ is even to obtain the uniform estimates of solutions in $$$ C\left(\left[0,T\right];{H}^s\left({\mathbb{R}}^d\right)\right) $$$ with respect to the Mach number. Compared with previous result, 19 for well‐prepared initial data, we remove this restriction and revisit the low Mach number limit in both the torus and the whole space $$$ {\mathbb{R}}^d $$$ under lower regularity assumptions by different approaches.…”

“…Recently, Liu and Xu 17 studied the incompressible limit in a bounded domain for well‐prepared initial data, and Zhang 18 proved the local well‐posedness and incompressible limit of the free‐boundary problem in the compressible elastodynamic equations. Recently, Wang 19 also studied the incompressible limit of nonisentropic Hookean elastodynamics for ill‐prepared initial data in the whole space $$$ {\mathbb{R}}^d $$$. One of the key assumptions in Wang 19 is that the index $$$ s\ge 4 $$$ is even to obtain the uniform estimates of solutions in $$$ C\left(\left[0,T\right];{H}^s\left({\mathbb{R}}^d\right)\right) $$$ with respect to the Mach number.…”

“…One of the key assumptions in Wang 19 is that the index $$$ s\ge 4 $$$ is even to obtain the uniform estimates of solutions in $$$ C\left(\left[0,T\right];{H}^s\left({\mathbb{R}}^d\right)\right) $$$ with respect to the Mach number. Compared with previous result, 19 for well‐prepared initial data, we remove this restriction and revisit the low Mach number limit in both the torus and the whole space $$$ {\mathbb{R}}^d $$$ under lower regularity assumptions by different approaches.…”