A Note on Strong Solutions to the Stokes System
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A Note on Strong Solutions to the Stokes System Luigi C. Berselli
Received: 31 October 2013 / Accepted: 14 February 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We give an alternative and quite simple proof of existence of W 2,q -W 1,q -strong solutions for the Stokes system, endowed with Dirichlet boundary conditions in a bounded smooth domain. Keywords Stokes system · Strong solutions · Existence Mathematics Subject Classification 35Q30
1 Introduction The aim of this note is to give a rather elementary proof of existence, uniqueness, and data dependence for strong solutions to the Stokes system with Dirichlet boundary conditions. Let Ω ⊂ R3 be an open bounded set with a C 1,1 boundary ∂Ω and let u = (u1 , u2 , u3 ) and π denote the unknown velocity and pressure, respectively. We will use customary Lebesgue Lq (Ω) and Sobolev W k,q (Ω) spaces (see e.g. Brezis [6]) and we define the 1,q following spaces: Vq := (W 2,q (Ω) ∩ W0 (Ω))3 , with norm uVq = uW 2,q , and also 1,q Mq := {f ∈ W (Ω) : Ω f dx = 0}, with norm f Mq = ∇f Lq , which is equivalent to that in W 1,q (Ω) thanks to the Poincaré inequality. We also set Xq := Vq × Mq . By adapting techniques from Beirão da Veiga [4], we will give an alternative proof of the following well-known result, fundamental in the theory of the Navier-Stokes equations.
Dedicated to Hugo Beirão da Veiga on the occasion of his 70th birthday.
B
L.C. Berselli ( ) Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, 56127, Pisa, Italy e-mail: [email protected] url: http://users.dma.unipi.it/berselli
L.C. Berselli
Theorem 1 Let be given 0 < ν ≤ 1, f ∈ (Lq (Ω))3 , and g ∈ Mq , for some 1 < q < ∞. Then, there exists a unique solution (u, π) ∈ Xq to the (non homogeneous) Stokes problem ⎧ ⎪ ⎨−νu + ∇π = f ∇ ·u=g ⎪ ⎩ u=0
in Ω, in Ω, on ∂Ω,
(1)
and there exists C = C(q, Ω) > 0 such that νuW 2,q + πW 1,q ≤ C(f Lq + gW 1,q ).
(2)
To prove Theorem 1 we will re-cast some standard tools in a new way. We recall that Theorem 1 dates back to Cattabriga [7] and to the announcement in Solonnikov [15] (the complete proofs of Solonnikov result appeared in the Russian version of Ladyžhenskaya book [13]), and the proofs are based on accurate analysis of hydrodynamic potentials. See also Vorovich and Yudovich [19] for q > 6/5; a different approach is presented in Amrouche and Girault [3], with the aid of vector potentials. As usual, the case q = 2 can be handled without potential theory, see Solonnikov and Šˇcadilov [17], Constantin and Foias [8], and a recent overview about the history of the problem can be also found in Galdi [9, Chap. IV]. Moreover, Beirão da Veiga [4] introduced for the case q = 2 an elegant, ingenious, and extremely simple approach, which is based essentially only on W 2,2 (Ω)-estimates for the solution of scalar Poisson problems. Inspired by the latter reference (especially [4, Sect. 4]) we will use the same technique to decouple the equation for the velocity from that for the pressure. Nevertheless, the extension t
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