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Existence and asymptotic behavior for strong solutions of the Navier- Stokes equations in the whole space. (English) Zbl 0601.35093
Consider the Navier-Stokes system of equations in the whole space \({\mathbb{R}}^ n\), \(n\geq 2\), namely: (1) \[ v'-\mu \Delta v+(v\cdot \nabla)v=f-\nabla p,\quad in\quad Q_ T\equiv]0,T[\times {\mathbb{R}}^ n, \]
\[ div v=0,\quad in\quad Q_ T,\quad v_{| t=0}=a(x)\quad in\quad {\mathbb{R}}^ n, \] where div f\(=0\) in \(Q_ T\), div a\(=0\) in \({\mathbb{R}}^ n\). A priori estimates and asymptotic behaviour as \(t\to +\infty\) are studied. Set \(L^{\alpha}=L^{\alpha}({\mathbb{R}}^ n)\), and denote \(| | _{\alpha}\) the norm in this space. A priori estimates for regular solutions, and lower bounds for the time of existence of regular solutions, are proved. In particular, some sharp results on global existence and asymptotic behaviour for strong solutions are proved. For the sake of brevity we only refer here to the last kind of results, and just in the particular case \(f\equiv 0:\) Let \(\alpha\) \(\geq n\) be fixed. There exist positive constants \(c_ 1,c_ 2,c_ 3\) (depending at most on \(\alpha\),n) such that if \[ | a| _ 2^{2(\alpha -n)/\alpha (n-2)}| a| _{\alpha}\leq c_ 1\mu ^{n(\alpha -2)//\alpha (n-2)}, \] then there exists a (unique) solution \(v\in L^ 2(0,+\infty;H^{1,2})\cap C(0,+\infty;L^{\alpha}\cap L^ 2)\) of system (1). Moreover, \[ | v(t)| _{\alpha}\leq | a| _{\alpha}[1+c_ 2\beta \mu | a| _ 2^{-\beta}| a| _{\alpha}^{\beta}t]^{-1/\beta}, \] \(\forall t\geq 0\), where \(\beta =4\alpha /(\alpha -2)n\). In particular (decay at infinity for the \(L^{\alpha}\)-norm) \[ | v(t)| _{\alpha}\leq c_ 3| a| _ 2(1/\mu t)^{(\alpha -2)n/4\alpha}. \] Note the particular form of the above expressions in case that \(\alpha =n\).

35Q30 Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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