On almost global existence for the Cauchy problem for compressible Navier-Stokes equations in the \(L_p\)-framework (Q2747068)

The Cauchy problem for the equation of compressible fluid motion is considered NEWLINE\[NEWLINE\begin{aligned} &\rho\left(\frac{\partial v}{\partial t}+ (v\cdot\nabla)v\right) -\mu\Delta v-\nu\nabla \text{div} v+\nabla p(\rho)=\rho f,\\&\frac{\partial \rho}{\partial t}+\text{div} (\rho v)=0, \qquad x\in \mathbb R^3,\;t>0,\\ &\rho|_{t=0}=\rho_0,\quad v|_{t=0}=v_0,\qquad x\in \mathbb R^3, \end{aligned}NEWLINE\]NEWLINEwhere \(v(x,t)\) is the velocity of the fluid, \(\rho (x,t)\) is the density, \(p(\rho)=a\rho^{\gamma}\) is the pressure, \(\mu\) and \(\nu\) are positive constants, \(\gamma>1\).NEWLINENEWLINENEWLINEIt is proved that for any \(T>0\) with sufficiently small \(v_0\) and \(\rho_0\) the problem has a regular solution \(v\in W^{2,1}_r(\mathbb R^3\times[0,T])\), \(\rho\in V_r(\mathbb R^3\times[0,T])\), where \(r>3\) and \(V_r(\mathbb R^3\times[0,T])\) is the closure of \(C^{\infty}(\mathbb R^3\times[0,T])\) in the norm NEWLINE\[NEWLINE\|u\|_{V_r(\mathbb R^3\times[0,T])}=\|u \|_{W^{1,0}_r(\mathbb R^3\times[0,T])} +\left\|\frac{\partial u}{\partial t}\right\|_{W^{1,0}_r(\mathbb R^3\times[0,T])}. NEWLINE\]NEWLINE The proof is based on \(L_r\)-estimates of solutions to the linearized equations.