Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Abstract

Abstract. The weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

“…Both of these problems have been overcome in Z07 [22] (see also Z08 [23]) for the case of the NavierStokes problem in a strip. The key idea there was to use the special weights ϕ ε (x) := (1 + ε 2 |x − x 0 |)…”

“…Then, the extra cubic term ϕ ′ ε u 3 ∼ εϕu 3 can be made small by the proper choice of the parameter ε depending on the "size" of the solution u. The second problem related with the pressure has been overcome by multiplying the equation by uϕ − v ϕ where v ϕ is a small corrector which makes this multiplier divergence free and which can be found as a solution of the proper linear conjugate equation, see Z07 [22] for the details. The main aim of the present paper is to extend the weighted energy method to the case of (damped) Navier-Stokes equations in the whole space.…”

“…The main aim of the present paper is to extend the weighted energy method to the case of (damped) Navier-Stokes equations in the whole space. Note that the analytic structure of the pressure term is essentially different than in the case of a strip with Dirichlet boundary conditions, so the method of Z07 [22] does not work in a direct way and requires an essential modification. In particular, we have to handle the pressure term in a completely different way based on the explicit formula for ∇p through u ⊗ u via the convolution operator and on the trick with the proper splitting of the convolution kernel developed in L02 [13] which allows us to treat ∇p in weighted and uniformly local spaces.…”

“…In particular, we have to handle the pressure term in a completely different way based on the explicit formula for ∇p through u ⊗ u via the convolution operator and on the trick with the proper splitting of the convolution kernel developed in L02 [13] which allows us to treat ∇p in weighted and uniformly local spaces. Thus, combining the ideas from Z07 [22] and L02 [13], we will give a comprehensive study of the damped Navier-Stokes equations in the uniformly local spaces.…”

“…The global in time bounds for such solutions are obtained combining the weighted energy estimates obtained in Section Finally, some approximations of divergence free vector which are necessary in order to prove the existence of solutions in uniformly local spaces are considered in Appendix 1 (see Section In this section, we briefly discuss the definitions and basic properties of the weighted and uniformly local Sobolev spaces (see MZ08,Z07,Z03 [14,22,24] for more detailed exposition). We start with the class of admissible weight functions and associated weighted spaces.…”