Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed

Abstract We consider the compressible Navier-Stokes-Korteweg equations describing the dynamics of a liquid-vapor mixture with diffuse interphase in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{d} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> d\ge 3 when the initial perturbation is suitably small. In particular, when the base sound speed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msqrt> <m:mrow> <m:mi>P</m:mi> <m:mo accent="false">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mrow> <m:mo>¯</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msqrt> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> \sqrt{P^{\prime} \left(\bar{\rho })}=0 , we first give the global existence and optimal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {L}^{2} -decay rate of the smooth solution, where the optimality means that the decay rate of the solution is the same as that for the corresponding linearized system, and there is no decay loss for the highest-order spatial derivatives of the solution. Then, we establish space-time behavior of the solution based on Green’s function method. It is obviously different from the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msqrt> <m:mrow> <m:mi>P</m:mi> <m:mo accent="false">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mrow> <m:mo>¯</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msqrt> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:math> \sqrt{P^{\prime} \left(\bar{\rho })}\gt 0 , which obeys the generalized Huygens’ principle as the compressible Navier-Stokes equations.