##
**On the existence of globally defined weak solutions to the Navier-Stokes equations.**
*(English)*
Zbl 0997.35043

The authors study the global existence of weak solutions to the Navier-Stokes equations of an isentropic compressible fluid:
\[
\rho_t+\text{div }(\rho\vec{u})=0,
\]

\[ (\rho u^i)_t+\text{div }(\rho u^i\vec{u})+a(\rho^{\gamma})_{x_i}=\mu\Delta u^i+(\lambda+\mu)(\text{div }\vec{u})_{x_i}, \] for \(0<t<T\) and \(x\in \Omega\) where \(\Omega\) is a bounded regular domain in \(\mathbb R^3\). Assume that the viscosity coefficients \(\lambda\) and \(\mu\) satisfy \(\mu>0\) and \(\lambda+\frac 23\mu\geq 0\), \(a>0\) and the adiabatic constant \(\gamma\) satisfies \(\gamma>3/2\).

The fixed initial conditions are: \( \rho(0)=\rho_0,\;(\rho u^i)(0)=q^i\) and \(u^i=0\) on \(\partial\Omega\).

The main theorem of the present paper states that, if \(\Omega\) is of class \(C^{2+\nu}\) with \(\nu>0\), and if the initial data satisfy the following compatibility conditions: \[ \rho_0\in L^{\gamma}(\Omega),\;\rho_0\geq 0,\;q^i(x)=0\text{ if }\rho_0(x)=0,\;\frac{\left|q^i\right|^2}{\rho_0}\in L^1(\Omega), \] then, for all \(T>0\), there exists a finite energy weak solution \(\rho,\vec{u}\) of the problem above, defined for \(0<t<T\). This generalizes a theorem by P. L. Lions [Mathematical topics in fluid dynamics, Vol 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications, Oxford (1998; Zbl 0908.76004)]. In particular, the case of monoatomic gas (\(\gamma=5/3\)) is included in the result of the present paper.

The first step of the proof consists in solving a modified problem with artificial viscosity and artificial pressure terms by means of a Faedo-Galerkin approximation. In the second step, one gets rid of the viscosity terms (using a technique developed by Lions, as well as a div-curl lemma), and finally, one gets rid of the pressure terms using cut-off operators introduced by the authors and others [Arch. Ration. Mech. Anal. 149, 69-96 (1999; Zbl 0937.35131)], [J. Differ. Equations 163, 57-75 (2000; Zbl 0952.35091)].

\[ (\rho u^i)_t+\text{div }(\rho u^i\vec{u})+a(\rho^{\gamma})_{x_i}=\mu\Delta u^i+(\lambda+\mu)(\text{div }\vec{u})_{x_i}, \] for \(0<t<T\) and \(x\in \Omega\) where \(\Omega\) is a bounded regular domain in \(\mathbb R^3\). Assume that the viscosity coefficients \(\lambda\) and \(\mu\) satisfy \(\mu>0\) and \(\lambda+\frac 23\mu\geq 0\), \(a>0\) and the adiabatic constant \(\gamma\) satisfies \(\gamma>3/2\).

The fixed initial conditions are: \( \rho(0)=\rho_0,\;(\rho u^i)(0)=q^i\) and \(u^i=0\) on \(\partial\Omega\).

The main theorem of the present paper states that, if \(\Omega\) is of class \(C^{2+\nu}\) with \(\nu>0\), and if the initial data satisfy the following compatibility conditions: \[ \rho_0\in L^{\gamma}(\Omega),\;\rho_0\geq 0,\;q^i(x)=0\text{ if }\rho_0(x)=0,\;\frac{\left|q^i\right|^2}{\rho_0}\in L^1(\Omega), \] then, for all \(T>0\), there exists a finite energy weak solution \(\rho,\vec{u}\) of the problem above, defined for \(0<t<T\). This generalizes a theorem by P. L. Lions [Mathematical topics in fluid dynamics, Vol 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications, Oxford (1998; Zbl 0908.76004)]. In particular, the case of monoatomic gas (\(\gamma=5/3\)) is included in the result of the present paper.

The first step of the proof consists in solving a modified problem with artificial viscosity and artificial pressure terms by means of a Faedo-Galerkin approximation. In the second step, one gets rid of the viscosity terms (using a technique developed by Lions, as well as a div-curl lemma), and finally, one gets rid of the pressure terms using cut-off operators introduced by the authors and others [Arch. Ration. Mech. Anal. 149, 69-96 (1999; Zbl 0937.35131)], [J. Differ. Equations 163, 57-75 (2000; Zbl 0952.35091)].

Reviewer: Emmanuel Russ (Marseille)

### MSC:

35Q30 | Navier-Stokes equations |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

35D05 | Existence of generalized solutions of PDE (MSC2000) |