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Two-dimensional Navier-Stokes flow with measures as initial vorticity. (English) Zbl 0666.76052
The main results of this work are some existence theorems for Navier- Stokes eqs. in velocity and pressure variables or vorticity variable when the initial vorticity is a finite measure in $$R^ 2.$$
The authors construct a global solution for these eqs. and prove regularity for $$t>0$$ as well as some decay estimates as $$t\to \infty$$. They consider only two-dimensional flow. Their main result may be understood as an example of existence of solutions for nonlinear parabolic equations with measures as initial data (solutions which may have infinite energy).
They compare their results with those of T. Kato [Proc. Symp. Pure Math. 45, Pt. 2, 1-7 (1986; Zbl 0598.35093)], G. Ponce [Commun. Partial Differ. Equations 11, 483-511 (1986; Zbl 0594.35077)] or R. J. DiPerna and A. J. Majda [ibid. 40, No.3, 301-345 (1987)]. Some of their results are extensions. However the theorem of existence for Euler eqs. is new for initial data of class $$L^ p(R^ 2)$$.
Reviewer: C.I.Gheorghiu

MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations 35K55 Nonlinear parabolic equations 35K99 Parabolic equations and parabolic systems
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References:
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