Existence of global smooth solutions for non-isentropic compressible Euler equations with time-dependent damping

In this paper, we consider the non-isentropic compressible Euler equations with time-dependent damping − α ( t ) u in one space dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>p</m:mi> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mi>α</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> casesₜ-uₓ=0, \\ uₜ+p(v,s)ₓ=-α (t)u, \\ sₜ=0. cases Under some conditions on the initial data and the entropy function, we prove some existence theorems of global smooth solutions for the Cauchy problem. At present, there seems to be no research on this kind of problem.