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Asymptotic analysis of the linearized Navier-Stokes equations in a channel. (English) Zbl 0832.35112

The authors treat a series of problems which on the one hand are interesting for their own sake and on the other hand may serve as preparation to more complex problems such as the following. Consider the Navier-Stokes equations (N-S): \[ u_t= \varepsilon\Delta u- (u\nabla) u- \nabla p+ f\quad\text{on }\Omega,\;u(\cdot, 0)= u_0,\;u= 0\quad\text{on }\partial \Omega\quad\text{and div } u= 0;\tag{1} \] let \(u^\varepsilon\), \(p^\varepsilon\) be the corresponding solution. Consider also the Euler equations: \[ u_t= -(u\nabla) u- \nabla p+ f\quad\text{on }\Omega,\;u(\cdot, 0)= u_0,\;\text{div } u= 0;\tag{2} \] let \(u^0\), \(p^0\) be the associated solution. The problem is: does \[ \text{does } \lim u^\varepsilon= u^0,\quad \lim p^\varepsilon= p^0\tag{3} \] hold in some sense as \(\varepsilon\to 0\)? The authors do not solve (3) but in order to get some insight into (3) they consider successively a one-dimensional heat equation on \((0, 1)\), a two- dimensional heat equation on the channel \(\mathbb{R}\times (0, 1)\) and finally the linearized N-S again on the channel \(\mathbb{R}\times (0, 1)\) from the above point of view. We briefly digress on the first of these equations, which is: \[ \partial_t u^\varepsilon= \varepsilon \partial^2_y u^\varepsilon+ f\quad\text{on }\Omega= (0, 1),\quad\text{with } u^\varepsilon(\cdot, 0)= u_0\quad\text{and } u(\cdot, t)= 0\quad\text{on }\partial \Omega.\tag{4} \] The limit case then is: \(\partial_t u^0= f\), \(u(\cdot, 0)= u_0\). One of the complications is that \(u_0\) must not satisfy \(u_0= 0\) on \(\partial\Omega\). The authors now define a corrector \(\theta^\varepsilon\) according to: \[ \partial_t \theta^\varepsilon= \varepsilon\partial^2_y \theta^\varepsilon,\;\theta^\varepsilon(\cdot, 0)= 0,\;\theta^\varepsilon(\cdot, t)= - u^0(\cdot, t)\quad\text{on } \partial\Omega. \] By energy estimates one shows that the corrector satisfies: \[ |u^\varepsilon- (u^0+ \theta^\varepsilon)|_T\leq k\varepsilon,\tag{5} \] where \(|\;|_T\) is the norm on \(L^2(0, T; L^2(\Omega))\), \[ |u^\varepsilon- (\theta^\varepsilon+ u^0)|_T\leq k\varepsilon^{{1\over 2}},\tag{6} \] where \(|\;|_T\) is the norm on \(L^2(0, T; H^1(\Omega))\). Here \(k= k(T, u^0, f)\) does not depend on \(\varepsilon\). The authors then show that (5), (6) still hold if \(\theta^\varepsilon\) is replaced by a more constructive corrector \(\widetilde\theta^\varepsilon\) whose definition cannot be reproduced here. A similar program is then carried out for the other two equations mentioned above. Finally, the large time behaviour of the solutions is discussed.

MSC:

35Q30 Navier-Stokes equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
76D07 Stokes and related (Oseen, etc.) flows
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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