Time decay and exponential stability of solutions to the periodic 3D Navier–Stokes equation in critical spaces

Abstract

Communicated by S. GeorgievUsing analysis in frequency space and Fourier methods, we establish that the global solution to the three-dimensional incompressible periodic Navier-Stokes equation for initial data in the critical Sobolev space P H 1=2 T 3 decays exponentially fast to zero, and it is exponentially stable as time goes to infinity as soon as the initial data (hence the solution) are mean free; otherwise, the difference to the average does so. Furthermore, we prove that any global nonmean-free solution vanishes as time goes to infinity, and it is globally exponentially stable. The main tools are the energy methods, the Friedrich's approximating schema, and a crucial change of function that depends explicitly on time.

Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

“…We observe that for the set K * of non-trivial resonant frequencies, Lemma 5.1 shows that Λ := (n, k, m) (−n, k, m) ∈ K * satisfies the conditions in Lemma 5.4 with any ρ ∈ (1,3]. We also notice that B R (a osc , a osc ) , a osc H 1 = B R (∇a osc , a osc ) , ∇a osc L 2 , since B R (a osc , ∇a osc ) , ∇a osc L 2 = 0 by Lemma 3.2.…”

“…Proof. We will prove it by constructing infinitely many triplets {(k j , m j , n j )} j≥1 satisfying k j,3 m j,3 n j,3 = 0, k j + m j = n j , n j ∈ L (1,0,1) and generating mutually different irreducible curves Γ(k j , m j , n j ) in the (θ 2 , θ 3 )-plane passing through (1,1).…”

“…So far, we have obtained a sequence of triplets {(k j , m j , n j )} j≥1 , k j = (a j , 1, a j+1 ), m j = (a j+1 , −1, a j ), n j = (a j + a j+1 , 0, a j + a j+1 ), for which k j + m j = n j , n j ∈ L (1,0,1) , and by the above construction of {a j }, the curve Γ(k j , m j , n j ) passes through (1,1). (Given the sequence {a j } defined by (4.12), one can also show directly without using the theory of Pell's equation that (x, y) = (a j , a j+1 ) satisfies (4.10), and hence (1, 1) ∈ Γ(k j , m j , n j ), by an induction on j.…”

Well‐posedness and convergence results for strong solution to a 3D‐regularized Boussinesq system

“…In the case of geophysical magnetohydrodynamic systems, we have used frequency analysis to deal with existence, uniqueness, and convergence results as a small parameter (Rossby number) vanishes [2][3][4][5][6]. Using the Fourier transform as a principal tool, we also gave an asymptotic study and stability results for both two-dimensional Leray weak solutions [7] and three-dimensional Fujita-Kato strong solution [8,9] to the periodic Navier-Stokes equation in critical spaces as time goes to infinity.…”

Study of the Blow Up of the Maximal Solution to the Three-Dimensional Magnetohydrodynamic System in Lei-Lin-Gevrey Spaces

“…Such assumption is mandatory to run the smallness argument and to obtain global well-posedness; see for example [11][12][13][14] and a complete survey in [10]. For many fluid mechanics equations, well-posedness and asymptotic behavior, as time goes to infinity or as small parameter goes to zero, were investigated by the authors, in various spaces; see for example [6][7][8][9][18][19][20]. About blowup, it is worthwhile to emphasize that several authors studied this phenomena to the Navier-Stokes equations; see for example [1][2][3]17] and references therein.…”