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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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[1] Leray J., J. Math. Pures Appl 12 pp 1– (1933)
[2] DOI: 10.1007/BF01474610 · Zbl 0008.06901 · doi:10.1007/BF01474610
[3] Golovkin K.K., Trud. Mat. Inst. Steklov 92 pp 33– (1968)
[4] DOI: 10.1007/BF00251436 · Zbl 0187.49508 · doi:10.1007/BF00251436
[5] T . Kato Remarks on the Euler and Navier-Strokes equations in R2,preprint
[6] G. Ponce,On persistent properties of the solutions to a class of nonlinear wave equations, preprint
[7] DOI: 10.1090/S0025-5718-1981-0628693-0 · doi:10.1090/S0025-5718-1981-0628693-0
[8] DOI: 10.1007/BF01174182 · Zbl 0545.35073 · doi:10.1007/BF01174182
[9] Bergh, J. and Löfström, J. 1970. ”Interpolation Spaces”. Berlin, New York; Springer · Zbl 0344.46071
[10] s. Klainerman, ”Nonlinear hyperbolic equations’. Lecture Notes , at Courant Institute (in Prepration). · Zbl 0648.53047
[11] DOI: 10.1002/cpa.3160340405 · Zbl 0476.76068 · doi:10.1002/cpa.3160340405
[12] G. Ponce , the initial value problem for there dimensional incompressible fluids ,preprint
[13] DOI: 10.2307/1970699 · Zbl 0211.57401 · doi:10.2307/1970699
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