## 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)].

### 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)

### Citations:

Zbl 0937.35131; Zbl 0952.35091; Zbl 0908.76004
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