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Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces $$L^ p_ s({\mathbb{R}}^ 2)$$. (English) Zbl 0615.35078
This paper is a sequel to the authors’ earlier papers [the first author, Proc. Symp. Pure Math. 45, Pt. 2, 1-7 (1986; Zbl 0598.35093) and the second author, Commun. Partial Differ. Equations 11, 483-511 (1986; Zbl 0594.35077)] and is concerned with the well-posedness (i.e., existence, uniqueness, and persistence) of the viscous and nonviscous solutions of the fluid dynamical equations as well as with the convergence of the viscous solutions as the viscosity tends to zero. It is shown that the Navier-Stokes (NS) and Euler equations for incompressible fluids possess well-posedness in the function space $$X=PL^ p_ s$$ with $$s>1+2/p$$, $$1<p<\infty$$ and P is a pseudo-differential operator.
In particular, starting from the basic initial value problem for the NS equations, the following results have been established.
(i) The NS equation in $${\mathbb{R}}^ m$$ is locally (in time) well-posed in the Lebesgue spaces $$PL^ p_ s({\mathbb{R}}^ m)$$, $$1<p<\infty$$, $$s>1+m/p.$$
(ii) The NS equation is globally well-posed in $$PL^ p_ s({\mathbb{R}}^ 2)$$ (p,s, as before) with uniform estimates for the viscosity $$\nu >0.$$
(iii) The Euler equation is globally well-posed in $$PL^ p_ s({\mathbb{R}}^ 2).$$
(iv) As $$\nu\to 0$$, the NS solution $$u^{\nu}(t)$$ converges to the Euler solution u(t) weakly in $$PL^ p_ s$$ for all $$t\geq 0$$, locally uniformly in t.
The results (ii)-(iv) have been established with the help of the vorticity equation derived from the Navier-Stokes equations and its properties.
Reviewer: P.Chandran

##### MSC:
 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 58D30 Applications of manifolds of mappings to the sciences 46N99 Miscellaneous applications of functional analysis 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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