An example of nonexistence globally in time of a solution of the Navier-Stokes equations for a compressible viscous barotropic fluid. (English. Russian original) Zbl 0877.35092

Russ. Acad. Sci., Dokl., Math. 50, No. 3, 397-399 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 339, No. 2, 155-156 (1994).
The author considers the initial-boundary value problem \[ \begin{gathered}\rho (\vec u_t +(\vec u\cdot\nabla)\vec u)-\mu\Delta\vec u -(\mu+\lambda)\nabla\text{div} \vec u+\nabla p=\rho\vec f,\\ \rho_t+\text{div}(\rho\vec u)=0,\qquad x\in\Omega,\quad t\in(0,T)\\ \rho|_{t=0}=\rho_0(x),\quad \vec u|_{t=0}=\vec u_0(x),\quad\vec u|_S=0 \end{gathered} \] in which the unknown functions are the density \(\rho(x,t)\), and the velocity vector \(\vec u(x,t)\) of a viscous compressible fluid filling a bounded domain \(\Omega\subset \mathbb R^n, n\geq 2\), with the boundary \(S\in C^k, k\geq 2\). The pressure \(p=R\rho^\gamma\), where \(R=\text{const}>0, \gamma=\text{const}\geq 0\), \(\lambda\) and \(\mu\) are positive constants. It was proved by V. A. Solonnikov [J. Sov. Math. 14, 1120–1133 (1980; Zbl 0451.35092); translation from Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov 56, 128–142 (1976; Zbl 0338.35078)] that the problem has a unique solution for \(T\leq T_0\) with small \(T_0\) and smooth data of the problem. The author constructs an example where the solution which exists for \(t\in (0,T_0]\) cannot be extended into the interval \((0,1)\).


35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics