On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains
The authors consider the ``stationary'' Navier-Stokes equations \[ \Delta\nu= \nu\cdot \nabla\nu+ \nabla p= f,\quad \nabla\cdot \nu= 0\tag{1} \] in the exterior \(\Omega\) of an obstacle \(B\), subject to the conditions \[ \nu(x)= \nu_*(x)\tag{2} \] for prescribed \(\nu_*(x)\) on \(\partial\Omega\), and \[ \lim_{x\to \infty} \nu(x)= \nu_\infty\tag{3} \] for prescribed \(\nu_\infty\). In \(n= 3\) dimensions, it is known that the problem admits a solution in the class \(D\) for which the Dirichlet integral of \(\nu\) is finite; this solution assumes all data strictly, and lies also in the class \(PR\) of ``physically reasonable'' solutions for which \(\nu(x)\) decays to \(\nu_\infty\) at prescribed rates that are more rapid in directions outside a paraboloidal ``wake'' region behind the obstacle. If \(n= 2\), the corresponding result has been proved only for ``small enough'' data. Without that restriction, existence is known of a solution in the class \(D\), for which (2) holds in a weak sense. In the present work, the authors prove for the case \(n= 2\) that if \(f\) has compact support, if (3) holds and (2) holds strictly for a solution \(\nu(x)\in D\) of (1), and if for some \(s> 1\) and \(\rho> 0\) there holds for the velocity component \(\nu_2\) orthogonal to \(\nu_\infty\), \[ \int_{|x|> \rho} |\nu_2(x)|^s dx< \infty\tag{4} \] then \(\nu(x)\) is in the class \(PR\). The result is of interest, in part because (4) is always satisfied for functions in \(PR\), and also because examples are known in \(n= 2\), of exterior flows whose components tend to limits at infinity arbitrarily slowly. The proof proceeds by delicate use of \(L^q\) estimates, to show that under the hypotheses every solution of (1) exhibits the same summability properties as the Oseen fundamental tensor for the equations (1) linearized about \(\nu_\infty\).
scientific article; zbMATH DE number 797097
scientific article