Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains
- PDF / 645,226 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 26 Downloads / 189 Views
Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains From the Weight Function Method to the Well Posedeness in L∞ and in Hölder Continuous Functional Spaces Paolo Maremonti
Received: 14 November 2013 / Accepted: 28 February 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361– 366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506– 511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37– 52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633– 652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011). Keywords Navier-Stokes equations · Non-decaying data · Well-posedeness 1 Introduction In this note, we prove a new result concerning the existence and uniqueness of non-decaying solutions to the Navier-Stokes initial boundary value problem in exterior domains: vt + v · ∇v + ∇πv = v, v=0
B
on (0, T ) × ∂Ω,
∇ · v = 0,
in (0, T ) × Ω,
v(0, x) = v◦ (x)
on {0} × Ω.
P. Maremonti ( ) Department of Mathematics and Physics, Seconda Università degli Studi di Napoli, via Vivaldi, 43, 81100 Caserta, Italy e-mail: [email protected]
(1)
P. Maremonti
In system (1) v is the kinetic field, πv is the pressure field, vt ≡ ∂t∂ v and v · ∇v ≡ vk ∂x∂ k v. For the sake of simplicity, we assume that the domain Ω is smooth, zero body force, and homogeneous boundary data. Our result completes the ones concerning the Cauchy problem [13, 19], the initial boundary value problem in the half-space [19] and the case of the exterior domain [8]. Actually, in [8] it is proved a result close to the one contained in this note. The difference is in the regularity of the initial data v◦ . Here, we assume v◦ ∈ L∞ (Ω) and ∇ ·v◦ = 0 in weak sense. Instead, in [8] it is assumed v◦ ∈ C 0,α (Ω) ∩ L∞ (Ω) and ∇ · v◦ = 0 in a weak sense. Following [1, 2], saying ∇ · v◦ = 0 in weak sense we mean that Ω v◦ · ∇ϕdx = 0, for 1,1 1,1 oc oc all ϕ ∈ W (Ω), where W (Ω) = {ϕ : ϕ ∈ L1oc (Ω) and ∇ϕ ∈ L1 (Ω)}. The details of our results are put behind the following brief review of the topic. Th
Data Loading...
