On the Well-Posedness of the Compressible Navier-Stokes-Korteweg System with Special Viscosity and Capillarity

Abstract

In this paper we consider the Cauchy problem to the compressible Navier-Stokes-Korteweg system with a specific choice on the viscosity and capillarity in the whole space ${\mathbb {R}}^d$, and establish the local well-posedness of strong solutions for large initial data in the framework of Besov spaces. Furthermore, we construct the global-in-time small solutions in $L^2$ type critical Besov spaces, where the vertical component of the divergence-free part of the velocity could be arbitrarily large initially.

Appendix

For the sake of convenience, here we derive the new system (1.5) by presenting more details in the computations.

Proof

Due to the mass equation, $\partial _t \rho +u\cdot \nabla \rho +\rho {\mathrm{div}}u=0$, one has

$$\begin{aligned} \partial _t\nabla \rho +u\cdot \nabla \nabla \rho +\nabla \rho \cdot \nabla ^{{\mathsf {T}}} u+\nabla \rho {\mathrm{div}}u+\rho \nabla {\mathrm{div}}u=0. \end{aligned}$$

(5.1)

From the momentum equations, one has

$$\begin{aligned}&\rho (\partial _t u+ u\cdot \nabla u-\mu \Delta u-\mu \nabla {\mathrm{div}}u)-2\mu \nabla \rho \cdot {\mathbb {D}}u\\&\quad -\lambda (\rho )\nabla {\mathrm{div}}u-\nabla \lambda (\rho ){\mathrm{div}}u+\nabla P(\rho ) =\kappa \rho \nabla \Delta \rho , \end{aligned}$$

hence, as long as $\rho$ does not vanish, which reduces to

$$\begin{aligned} \partial _t u+ u\cdot \nabla u&-\mu \Delta u-(\mu +\rho ^{-1}\lambda (\rho )\nabla {\mathrm{div}}u\nonumber \\&-2\mu \nabla \ln \rho \cdot {\mathbb {D}}u-\rho ^{-1}\nabla \lambda (\rho ){\mathrm{div}}u+\rho ^{-1}\nabla P(\rho )=\kappa \nabla \Delta \rho . \end{aligned}$$

(5.2)

Introducing the effective velocity

$$\begin{aligned} {{\textbf {v}}}=u+\lambda \nabla \rho \end{aligned}$$

and adding the equality (5.2) to (5.1)$\times \lambda$ yields

$$\begin{aligned}&\partial _t {{\textbf {v}}}+ u\cdot \nabla {{\textbf {v}}}-\mu \Delta {{\textbf {v}}}+(\lambda \rho -\mu -\rho ^{-1}\lambda (\rho ))\nabla {\mathrm{div}}{{\textbf {v}}}-2\mu \nabla \ln \rho \cdot {\mathbb {D}}u+\lambda \nabla \rho \cdot \nabla ^{{\mathsf {T}}} u\nonumber \\&\quad \quad +(\lambda \nabla \rho -\rho ^{-1}\nabla \lambda (\rho )){\mathrm{div}}u+\rho ^{-1}\nabla P(\rho )\nonumber \\&\quad =\Big (\kappa -\lambda (2\mu +\rho ^{-1}\lambda (\rho )-\lambda \rho )\Big )\nabla \Delta \rho . \end{aligned}$$

(5.3)

In order to “kill” the third-order derivative term $\kappa \nabla \Delta \rho$, it is natural to assume the algebraic relation

$$\begin{aligned} \lambda (2\mu +\rho ^{-1}\lambda (\rho )-\lambda \rho )=\kappa , \end{aligned}$$

that is

$$\begin{aligned} \lambda (\rho )=\lambda \rho ^2+({\bar{\mu }}-2\mu )\rho \quad \text{ with }\quad {\bar{\mu }}=\frac{\kappa }{\lambda }, \end{aligned}$$

then we find from (5.2) that

$$\begin{aligned} \partial _t {{\textbf {v}}}+ u\cdot \nabla {{\textbf {v}}}&-\mu \Delta {{\textbf {v}}}-({\bar{\mu }}-\mu )\nabla {\mathrm{div}}{{\textbf {v}}}-2\mu \nabla \ln \rho \cdot {\mathbb {D}}u\nonumber \\&-({\bar{\mu }}-2\mu )\nabla \ln \rho {\mathrm{div}}u+\lambda \nabla \rho \cdot \nabla ^{{\mathsf {T}}} u\nonumber \\&-\lambda \nabla \rho {\mathrm{div}}u+\rho ^{-1}\nabla P(\rho )=0. \end{aligned}$$

(5.4)

To avoid the presence of u, notice the fact $u={{\textbf {v}}}-\lambda \nabla \rho$, we deduce

$$\begin{aligned} \partial _t {{\textbf {v}}}+ {{\textbf {v}}}\cdot \nabla {{\textbf {v}}}&-\mu \Delta {{\textbf {v}}}-({\bar{\mu }}-\mu )\nabla {\mathrm{div}}{{\textbf {v}}}+\rho ^{-1}\nabla P(\rho )\nonumber \\&=\lambda \nabla \rho \cdot \nabla {{\textbf {v}}}+\lambda \nabla \rho {\mathrm{div}}u-\lambda \nabla \rho \cdot \nabla ^{{\mathsf {T}}} u+2\mu \nabla \ln \rho \cdot {\mathbb {D}}u\nonumber \\&\quad +({\bar{\mu }}-2\mu )\nabla \ln \rho {\mathrm{div}}u\nonumber \\&=\lambda \nabla \rho \cdot \nabla {{\textbf {v}}}+\lambda \nabla \rho {\mathrm{div}}({{\textbf {v}}}-\lambda \nabla \rho )-\lambda \nabla \rho \cdot \nabla ^{{\mathsf {T}}} ({{\textbf {v}}}-\lambda \nabla \rho )\nonumber \\&\quad +2\mu \nabla \ln \rho \cdot {\mathbb {D}}({{\textbf {v}}}-\lambda \nabla \rho )\nonumber \\&\quad +({\bar{\mu }}-2\mu )\nabla \ln \rho {\mathrm{div}}({{\textbf {v}}}-\lambda \nabla \rho )\nonumber \\&=\nabla \rho \cdot [\lambda (\nabla -\nabla ^{{\mathsf {T}}}+{\mathbb {I}}{\mathrm{div}}){{\textbf {v}}}+\lambda ^2(\nabla ^2 -{\mathbb {I}}\Delta )\rho ]\nonumber \\&\quad +\nabla \ln \rho \cdot [(2\mu {\mathbb {D}}+({\bar{\mu }}-2\mu ){\mathbb {I}}{\mathrm{div}}){{\textbf {v}}}-\lambda (2\mu {\mathbb {D}}+({\bar{\mu }}-2\mu ){\mathbb {I}}{\mathrm{div}})\nabla \rho ]. \end{aligned}$$

(5.5)

Denoting $a=\rho -1$, we can reformulate the system (1.3) equivalently as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t a -\lambda (1+a)\Delta a+{\mathrm{div}}{{\textbf {v}}}-\lambda \nabla a\otimes \nabla a+{\mathrm{div}}(a {{\textbf {v}}})=0,\\ \partial _t {{\textbf {v}}}+ {{\textbf {v}}}\cdot \nabla {{\textbf {v}}}-\mu \Delta {{\textbf {v}}}-({\bar{\mu }}-\mu )\nabla {\mathrm{div}}{{\textbf {v}}}\\ \quad =\nabla a\cdot [\lambda (\nabla -\nabla ^{{\mathsf {T}}}+{\mathbb {I}}{\mathrm{div}}){{\textbf {v}}}+\lambda ^2(\nabla ^2 -{\mathbb {I}}\Delta )a] \\ \qquad +\nabla \ln \rho \cdot [(2\mu {\mathbb {D}}+({\bar{\mu }}-2\mu ){\mathbb {I}}{\mathrm{div}}){{\textbf {v}}}\\ \qquad -\lambda (2\mu {\mathbb {D}}+({\bar{\mu }}-2\mu ){\mathbb {I}}{\mathrm{div}})\nabla a]+ k(a)\nabla a-\nabla a,\\ ({{\textbf {v}}},a)|_{t=0}=({{\textbf {v}}}_{0},a_{0}).\end{array}\right. \end{aligned}$$

where we denote $k(s)=1-\frac{P'(1+s)}{1+s}$. $\square$