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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

MSC:
 35Q30 Navier-Stokes equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:
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