Global Well-Posedness for the fractional Navier-Stokes-Coriolis equations in Function spaces Characterized by Semigroups

We studies the initial value problem for the fractional Navier-Stokes-Coriolis equations, 1 which obtained by replacing the Laplacian operator in the Navier-Stokes-Coriolis equation by 2 the more general operator (−∆)α with α > 0. We introduce function spaces of the Besove type 3 characterized by the time evolution semigroup associated with the general linear Stokes-Coriolis 4 operator. Next, we establish the unique existence of global in time mild solutions for small initial 5 data belonging to our function spaces characterized by semigroups in both the scaling subcritical 6 and critical settings. 7


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In this paper, we mainly consider the case α > 0, i.e., prove the global well-38 posedness of (1) in some function spaces characterized by the time evolution semi-39 group generated by the linear operator (−∆) α + ΩPe 3 × P. Where P is the Helmholtz 40 projection.

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To study the problem (1), we consider the following equivalent integral equation where P = (δ ij + R i R j ) 1≤i,j≤3 denotes the Helmholtz projection onto the divergence free vector fields, T Ω (t) = e t(−(−∆) α −ΩPe 3 ×P) denotes the semigroup associated with linearized problem of (1), which is given explicitly by for t ≥ 0 and divergence-free vector field f . Here, I is the identity matrix in R 3 , R(ξ) is the skew-symmetric matrix defined by We refer to [1,3,11] for the derivation of the explicit form of T Ω (t) f . We say that u is

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(i) For s ∈ R and 1 ≤ p, θ ≤ ∞, the function space X s,p,θ Ω (R 3 ) is defined as follows: is defined as follows: Then, for 1 ≤ θ < ∞, according to [22], there is We also see that for Therefore, the function space X s,p,θ Ω (R 3 ) and X p Ω (R 3 ) can be regard as one of the generalizations 49 of the Besove spaceḂ

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(ii) For 3 ≤ p < ∞, we set 1 q := 1 p + 1 6 . We see that 2 ≤ q < 6 and the Sobolev embedding holds. Therefore, it follows from the Plancherel theorem and Lemma 6 that for all t > 0 and Ω ∈ R. Hence the continuous embeddingḢ In this paper, we show the unique existence of global in time solutions in the 55 subcritical spaces X s,p,θ The following is our result in scaling subcritical cases. 59 Theorem 1. Let α, s, p, and θ satisfy (9) Then, there exists positive constants C = C(α, s, p, θ) such that for Ω ∈ R \ {0} and for initial velocity u 0 ∈ X s,p,θ (2) possesses a unique mild solution In the case Ω = 0, as we have seen in (5), there hold for dyadic number λ > 0. Since 2α θ + 3 p < s + (2α − 1) by our assumption (9) we generalize their results for α = 1.

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Our main result in the scaling critical case reads as follows: 66 Theorem 2. Let α, s, p satisfy Then, there exists a positive constant δ = δ(α, s, p, ) independent of Ω ∈ R such that for the initial velocity (1) possesses a unique mild solution  The rest of this paper is organized as follows. In Section 2, we collect some basic 70 facts on Littlewood-Paley theory, and show some new linear estimates for semigroup 71 {T Ω (t)} t≥0 . In Section 3, we establish the bilinear estimates for the Duhamel terms in 72 (4). Finally, we present the proofs of the main results.

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Now, we introduce the definitions of homogeneous Besov spaceḂ s p,q (R 3 ).
Then, we can rewrite the operator T Ω (t) as for all t ≥ 0, where R denotes the matrix of singular integral operators defined by First, we recall the behavior of the fractional order heat semigroup e −t(−∆) α in 78 Lebesgue spaces.
By combing this with the Plancherel theorem, we obtain the linear estimates for the 87 semigroup T Ω (t).

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Proof. Since R is bounded in L 2 (R 3 ), it follows from the Plancherel theorem, L p 0 − L 2 90 estimate for the semigroup e −t(−∆) α , Lemma 3 and Lemma 4 that Thus, we complete the Proof of Lemma 5.

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By Lemma 4, the continuous embeddings ), we obtain the following Lemma 6.
. So, by the Hardy-Littlewood-Sobolev inequality , we see T for all Ω ∈ R. This completes the proof of Lemma 2.9.

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Next, we prove the inequality (19). Since 1 < p 2 ≤ 2, it follows from Lemma 6, the 121 Hölder inequality and the definitions of ∥ · ∥ Z 2 that where we remark that 1 2α + 3 2αp < 1 since p > 3 2α−1 and C independent of t. Thus, we 123 complete the proof of the inequality (19). ∥T Ω (·)u 0 ∥ Z 1 = ∥u 0 ∥ X s,p,θ Ω . Then, we define the complete metric space (X 1 , d 1 ) and the 129 map ψ by 130 where N(u, v)(t) is defined in (16). By the inequality (17), there exists a positive constant for all u ∈ X 1 . Moreover, by using inequality (17), there exists a positive constant C 1 133 such that for u, v ∈ X 1 , Now, let us assume that initial velocity u 0 ∈ X s,p,θ we obtain from (4.1) and (4.2) that for u, v ∈ X 1 . Therefore, by the contraction mapping principle, there exists a unique 135 solution u ∈ X 1 satisfying (3) for all t > 0.