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On two dimensional incompressible fluids. (English) Zbl 0594.35077
For the initial value problem (I.V.P) \[ (1)\quad u^ k_ t+(u.\nabla)u^ k=\nu \Delta u^ k-\partial_ kP,\quad x\in {\mathbb{R}}^ 2,\quad t\geq 0,\quad div u=0,\quad u(x,0)=u_ 0(x),\quad k=1,2, \] where \(u(x,t)=(u^ 1(x,t),u^ 2(x,t))\) is the velocity field, \(P=P(x,t)\) is the pressure, \(\nu\geq 0\), the following theorem is proven: Theorem. For any \(u_ 0\in L^ 2(R^ 2)\cap H^ s_ p(R^ 2)\), div \(u_ 0=0\), with \((s,p)\in [2,\infty]\times [2,\infty)\) (except for the case \((S,p)=(2,2))\) the I.V.P (1) has a unique global classical solution u, where \(u\in C([0,\infty):\quad L^ 2(R^ 2)\cap H^ s_ p(R^ 2)).\) Moreover for any \(T>0\), \(\sup_{[0,T]}\| u(.,t)\|_{H^ s_ p}\leq K\) where the constant K depends only on T, \(\| u_ 0\|_ 0\) and \(\| u_ 0\|_{H^ s_ p}\) but not on \(\nu\).
Reviewer: V.Kostova

35Q30 Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
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