Abstract
We study the Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier-Stokes and Euler type problems. In addition, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained.
Highlights
We study the system of Euler equations with the so-called Ekman damping in the whole 2D space
The weak attractors are normally used in order to describe the longtime behavior of solutions of the damped Euler equations
1 L∞ + α curl g L∞, ω := curl u, (1.2). This gives only growing in time estimates for the velocity u, see [14,28] even in a more simple case of damped Navier–Stokes equations, so in order to get the dissipative bounds for the velocity field, we need to use the energy type estimates
Summary
We briefly discuss the definitions and basic properties of the weighted and uniformly local Sobolev spaces (see [23,33,35] for more detailed exposition). The associated weighted Lebesgue space Lpφ(R2), 1 ≤ p < ∞, is defined as a subspace of functions belonging to Lploc(R2) for which the following norm is finite:. Φ(x) := e−ε|x−x0| or φ(x) := e− 1+ε2|x−x0|2 , ε ∈ R, x0 ∈ R2 Another class of admissible weights of exponential growth rate are the so-called polynomial weights and, in particular, the weight function. The space Lpb (R2) is defined as the subspace of functions of Lploc(R2) for which the following norm is finite: u. The proposition gives the useful equivalent norms in the weighted Sobolev spaces. Let φ be the weight of exponential growth rate such that x∈R2 φ dx < ∞. ≤C where C is independent of y0 ∈ R2 The proof of this corollary can be found in [36]. Wbl, where the constants C and C1 are independent of R 1
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