Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 89, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOGARITHMICALLY IMPROVED REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS IN HOMOGENEOUS BESOV SPACES NGUYEN ANH DAO, JESUS ILDEFONSO DÍAZ Abstract. We investigate a logarithmically improved regularity criteria in terms of the velocity, or the vorticity, for the Navier-Stokes equations in ho- mogeneous Besov spaces. More precisely, we prove that if the weak solution u satisfies either ∫ T 0 ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖u(t)‖Ḣs0 dt < ∞, or ∫ T 0 ‖w(t)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(t)‖Ḣs0 dt < ∞ , where w = rotu, then u is regular on (0, T ]. Our conclusions improve some results by Fan et al. [5]. 1. Introduction Our main purpose is to investigate a logarithmically improved regularity criteria of solutions to the Navier-Stokes equations in Rn, n ≥ 3: ∂tu−∆u+ u · ∇u+∇p = 0, x ∈ Rn, t ∈ (0, T ), div u = 0, u(x, 0) = u0(x), (1.1) where u(x, t) = (u1(x, t), . . . , un(x, t)), and p denote the velocity vector and pres- sure, respectively, of the fluid at the point (x, t) ∈ Rn × (0, T ) and u0 is a given initial velocity. Since the pioneering works by Leray [17] and Hopf [8], the existence of global weak solutions for an arbitrary initial data u0 ∈ L2(Rn) was well-known. However, the uniqueness and regularity of weak solutions are still open. Notice that the studying of blow-up of solutions to (1.1) plays a crucial role not only in nonlinear analysis, but also in the study of the regularity of weak solutions. Also, it is known that for each regular u0, there exists t0 > 0 such that u is regular for 0 ≤ t ≤ t0 (see e.g. [16]). Different regularity criteria for the weak solutions have been proposed. For example, the Prodi-Serrin conditions [22, 23] states that if the weak solution u satisfies u ∈ Lr(0, T ;Lp(Rn)) with 2 r + n p ≤ 1, n < p <∞, 2 < r <∞, 2010 Mathematics Subject Classification. 35Q35, 35B65, 76D05. Key words and phrases. Besov space; Navier-Stokes equations; regularity criteria. ©2021. This work is licensed under a CC BY 4.0 license. Submitted August 3, 2021. Published November 3, 2021. 1 2 N. A. DAO, J. I. DÍAZ EJDE-2021/89 then u is smooth on Rn × (0, T ). The limiting case, p = n and r = ∞, was obtained in Escauriaza et al. [4]. A logarithmically improved regularity criterion was introduced by Montgomery-Smith [18]. That is, if∫ T 0 ‖u(t)‖rLp 1 + log+ ‖u(t)‖Lp dt <∞, 2 r + 3 p = 1, 3 < p <∞, then u is smooth (in the sense that u is at least in the Sobolev spaces Wn,q for some q ∈ [2,+∞) and all positive integers n). On the other hand, in 1995, Beirão da Veiga [2] established a Serrin’s type regularity criterion on the gradient of velocity field: ∇u ∈ Lr(0, T ;Lp(Rn)) with 2 r + n p ≤ 2. Beale–Kato–Majda [1] and Kato–Ponce [11] showed that the L∞-norm of the vorticity, denoted by w = rotu, controls the breakdown of smooth solutions to the Euler and Navier-Stokes equations. To be more precise, if∫ T 0 ‖w(τ)‖L∞ dτ <∞ , then the smooth solution u, in C([0, T );W s,p(Rn)), with s > n/p + 1, can be continued beyond t = T . That was improved by Kozono–Taniuchi [14, 15] in BMO(Rn). Theorem 1.1. Let s > n 2 − 1 and let u0 ∈ Hs(Rn) with div u0 = 0. Suppose that u is a strong solution of (1.1) in the class ST := C ((0, T );Hs(Rn)) ∩ C1 ((0, T );Hs(Rn)) ∩ C ( (0, T );Hs+2(Rn) ) . If ∫ T δ0 ‖w(τ)‖BMO dτ <∞ , (1.2) for some δ0 ∈ (0, T ), then u can be continued as solution in the class ST ′ for some T ′ > T . In addition, Kozono et al. [12] improved Theorem 1.1 in the homogeneneous Besov space (for the definition of this and other spaces we will mention in this Introduction we send, for instance, to the exposition made in the monographs [24] and [16]): ∫ T 0 ‖w(τ)‖Ḃ0 ∞,∞ dτ <∞ . (1.3) A version of this, as a logarithmically improved regularity criterion of (1.3), was given in Fan et al. [6]: ∫ T 0 ‖w(τ)‖Ḃ0 ∞,∞√ 1 + log+ ‖w(τ)‖Ḃ0 ∞,∞ dτ <∞ . (1.4) Recently, Nakao-Taniuchi [19] proved a different logarithmically improved regularity criterion as follows: ∫ T 0 ‖w(τ)‖BMO 1 + log+ ‖u(τ)‖C1+α dτ <∞ (1.5) for some α ∈ (0, 1). We point out that these authors obtained (1.4) by using the Brézis–Gallouët-Wainger type inequality. Concerning the logarithmically improved regularity criterion on the homoge- neous Besov space Ḃ−α∞,∞, Fan et al. [5] proved the following results. EJDE-2021/89 LOGARITHMICALLY IMPROVED REGULARITY CRITERIA 3 Theorem 1.2 ([5]). Let u0 ∈ L2n(Rn) with div u0 = 0. Let u ∈ L∞ ( 0, T ;L2(Rn) ) ∩ L2 ( 0, T ;H1(Rn) ) be a weak solution of (1.1). Assume that one of the following conditions is satisfied: ∫ T 0 ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖u(t)‖Ḃ−α∞,∞ dt <∞, with 0 < α < 1 , (1.6) ∫ T 0 ‖w(t)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(t)‖Ḃ−α∞,∞ dt <∞, with n = 3, and 0 < α < 1 . (1.7) Then u is smooth on (0, T ]. The main goal of this paper is to improve (1.6) and (1.7). Our main results read as follows. Theorem 1.3. Let u0 ∈ L2n(Rn) with div u0 = 0. Let u ∈ L∞ ( 0, T ;L2(Rn) ) ∩ L2 ( 0, T ;H1(Rn) ) be a weak solution of (1.1). Suppose that ∫ T 0 ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖u(t)‖Ḣs0 dt <∞ (1.8) holds for some α ∈ (0, 1), with s0 = n 2 − α. Then u is smooth on (0, T ]. As a consequence of the above theorem and the Sobolev embedding, we have the following corollary. Corollary 1.4. Let u0 ∈ L2n(Rn) with div u0 = 0. Let u ∈ L∞ ( 0, T ;L2(Rn) ) ∩ L2 ( 0, T ;H1(Rn) ) be a weak solution of (1.1). If in addition ∫ T 0 ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖u(t)‖Ln/α dt <∞ , (1.9) for some α ∈ (0, 1), then u is smooth on (0, T ]. Remark 1.5. It is clear that (1.6) is weaker than (1.9) (see Proposition 2.4 below). Our last result in this paper improves condition (1.7). Theorem 1.6. Let u0 ∈ L6(R3) with div u0 = 0. Let u ∈ L∞ ( 0, T ;L2(R3) ) ∩ L2 ( 0, T ;H1(R3) ) be a weak solution of (1.1). If ∫ T 0 ‖w(t)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(t)‖Ḣs0 dt <∞ (1.10) for 0 < α < 1, s0 = 3 2 − α, then u is smooth on (0, T ]. Notation. Through this paper, we use the following general abbreviation X = X(Rn). So, for instance, Lp ≡ Lp(Rn), and Hs ≡ Hs(Rn). Moreover, we denote by C a positive constant which can change from line to line. 4 N. A. DAO, J. I. DÍAZ EJDE-2021/89 2. Definitions and preliminary results Let us first define a weak solution, introduced by Leray [17]. Definition 2.1. Let u0 ∈ L2 with div u0 = 0 in Rn. Then u is called a weak solution of (1.1) if u ∈ L∞ ( 0, T ;L2(Rn) ) ∩L2 ( 0, T ;H1(Rn) ) satisfies the equation in distributional sense and the following inequality ‖u(t)‖2L2 + 2 ∫ t 0 ‖∇u(τ)‖2L2 dτ ≤ ‖u0‖2L2 (2.1) for all t ∈ (0, T ). The following results (see e.g. [10, Theorem 4], [7]) will be repeatedly used. Proposition 2.2. (i) Suppose that u0 ∈ Lγ , for some γ ≥ n with div u0 = 0 in Rn. Then, there exists a time T0 > 0 and a unique solution of (1.1) on [0, T0) such that u ∈ BC ([0, T0);Lγ) ∩ Ls (0, T0;Lr) , t1/su ∈ BC ([0, T0);Lr) , (2.2) with 2 s + n r = n γ , s, r > n, and BC denotes the space of bounded and continuous functions. (ii) Let (0, T ∗) be the maximal interval such that u solves (1.1) in C ((0, T ∗), Lγ), with γ > n. Then ‖u(t)‖Lγ ≥ C(T ∗ − t) n−γ 2γ , (2.3) where constant C > 0 is independent of T ∗ and t. (iii) Let u be a solution of (1.1) on (0, T0) in the functions class (2.2). Suppose that u0 ∈ L2. Then u is also a weak solution in Definition 2.1. (iv) Let u be a weak solution of (1.1) satisfying u ∈ Ls (0, T ;Lr(Rn)), for some r > n, with 2 s + n r ≤ 1. Then u ∈ C∞ (Rn × (0, T )). To define the homogeneous Besov spaces, we recall the Littlewood-Paley decom- position (see, e.g., [24]). Let φj(x) be the inverse Fourier transform of the j-th component of the dyadic decomposition i.e.,∑ j∈Z φ̂(2−jξ) = 1 except ξ = 0, where supp(φ̂) ⊂ {ξ : 1/2 < |ξ| < 2}. Let Z(Rn) = { f ∈ S(Rn), Dαf̂(0) = 0, ∀α ∈ Nn, multi-index } . Definition 2.3. For every s ∈ R, and for every 1 ≤ q, r ≤ ∞, the homogeneous Besov space is denoted by Ḃsq,r = { f ∈ Z ′(Rn) : ‖f‖Ḃsq,r <∞ } , with ‖f‖Ḃsq,r = {(∑ j∈Z 2jsr‖φj ∗ f‖rLq )1/r , if 1 ≤ r <∞, supj∈Z{2js‖φj ∗ f‖Lq} , if r =∞ , where φj(x) = 2jnφ(2jx). Proposition 2.4. For any 0 < σ ≤ n, we have L n σ (Rn) ↪→ Ḃ−σ∞,∞(Rn) . EJDE-2021/89 LOGARITHMICALLY IMPROVED REGULARITY CRITERIA 5 Proof. From Young’s inequality, for any j ∈ Z, we have 2−jσ‖φj ∗ f‖L∞ ≤ C2−jσ‖φj‖L n n−σ ‖f‖Lnσ ≤ C‖f‖Lnσ . This completes the proof. � 3. Proof of main results Proof of Theorem 1.3. Since u ∈ L2n, by applying Proposition 2.2, we obtain a weak solution u which is smooth in (0, T0). Therefore, for any T > 0, we can assume that u is smooth on (0, T ). For s ∈ (s0, n 2 ), applying (−∆)s/2 to (1.1), and using (−∆)s/2u as a test function to the resulting equation we obtain 1 2 d dt ∫ |(−∆)s/2u(t)|2 dx+ ∫ |∇(−∆)s/2u|2 dx = − ∫ (−∆)s/2(u · ∇u) · (−∆)s/2u dx = − ∫ (−∆)s/2 div(u⊗ u) · (−∆)s/2u dx = − ∫ (−∆) s−α 2 div(u⊗ u) · (−∆) s+α 2 u dx ≤ ‖u⊗ u‖H1+s−α‖(−∆) s+α 2 u‖L2 ≤ ‖u‖Ḃ−α∞,∞‖(−∆) s+1 2 u‖L2‖(−∆)s/2u‖1−αL2 ‖(−∆) s+1 2 u‖αL2 ≤ δ‖(−∆) s+1 2 u‖2L2 + Cδ‖u‖ 2 1−α Ḃ−α∞,∞ ‖(−∆)s/2u‖2L2 , where δ > 0 is small enough. Notice that we have used the inequality ([13]) ‖u⊗ u‖H1+s−α ≤ C‖u‖Ḃ−α∞,∞‖(−∆) s+1 2 u‖L2 , the Gagliardo–Nirenberg inequality [3], and the Young inequality. Therefore, 1 2 d dt ‖(−∆)s/2u(t)‖2L2 ≤ C‖u(t)‖ 2 1−α Ḃ−α∞,∞ ‖(−∆)s/2u(t)‖2L2 ≤ C ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖(−∆) s0 2 u(t)‖L2 ‖(−∆)s/2u(t)‖2L2 × ( 1 + log+ ‖(−∆)s0/2u(t)‖L2 ) . (3.1) Thanks to the Gagliardo–Nirenberg inequality, we obtain ‖(−∆) s0 2 u(t)‖L2 ≤ ‖u(t)‖1− s0 s L2 ‖(−∆)s/2u(t)‖ s0 s L2 ≤ ‖u‖ 1− s0s L∞(0,T ;L2)‖(−∆)s/2u(t)‖ s0 s L2 for t ∈ (0, T ). Since u ∈ L∞ ( 0, T ;L2(Rn) ) , it follows from the above inequality that 1 + log+ ‖(−∆) s0 2 u(t)‖L2 ≤ C log ( e+ ‖(−∆)s/2u(t)‖2L2 ) . (3.2) Combining (3.1) and (3.2) we obtain d dt ‖(−∆)s/2u(t)‖2L2 6 N. A. DAO, J. I. DÍAZ EJDE-2021/89 ≤ C ‖u(t)‖ 2 1−α Ḃ−α∞,∞ 1 + log+ ‖(−∆) s0 2 u(t)‖L2 ‖(−∆)s/2u(t)‖L2 log ( e+ ‖(−∆)s/2u(t)‖2L2 ) , which implies ‖u‖L∞(0,T ;Hs) ≤ C . Hence, from the Sobolev embedding, we deduce that u ∈ L∞(0, T ;L 2n n−2s (Rn)) . This and Proposition 2.2 imply that u is smooth on [0, T ]. The proof of Theorem 1.3 is complete. � Proof of Theorem 1.6. It is not difficult to verify that w satisfies the equation wt + u · ∇w −∆w = w · ∇u . (3.3) Testing (3.3) with −∆w, and using that div u = 0 we obtain 1 2 d dt ‖∇w(t)‖2L2 + ‖∆w(t)‖2L2 = ∫ (u · ∇)w ·∆w dx− ∫ (w · ∇)u ·∆w dx = ∑ i,j ∫ ui∂iw · ∂2 jw dx+ ∫ (−∆) 1−α 2 (w · ∇u) · (−∆) 1+α 2 w dx = − ∑ i,j ∫ ∂jui∂iw · ∂jw dx+ ∫ (−∆) 1−α 2 (w · ∇u) · (−∆) 1+α 2 w dx = − ∑ i,j ∫ ∂i(∂juiw) · ∂jw dx+ ∫ (−∆) 1−α 2 (w · ∇u) · (−∆) 1+α 2 w dx = − ∑ i,j ∫ (−∆) −α 2 ∂i(∂juiw) · (−∆) α 2 ∂jw dx + ∫ (−∆) 1−α 2 (w · ∇u) · (−∆) 1+α 2 w dx . (3.4) By Hölder’s inequality and the Plancherel theorem, we obtain∣∣∣ ∫ (−∆) −α 2 ∂i(∂juiw) · (−∆) α 2 ∂jw dx+ ∫ (−∆) 1−α 2 (w · ∇u) · (−∆) 1+α 2 w dx ∣∣∣ ≤ C‖(−∆) 1−α 2 (w · ∇u)‖L2‖(−∆) 1+α 2 (w)‖L2 . (3.5) On the other hand, we recall the following two inequalities obtained in [13] and [9]: ‖fg‖Ḣs ≤ C ( ‖f‖Ḃ−α∞,∞‖g‖Ḣs+α + ‖g‖Ḃ−α∞,∞‖f‖Ḣs+α ) , ‖∇u‖Ḃ−α∞,∞ ≤ C‖w‖Ḃ−α∞,∞ . Then ‖(−∆) 1−α 2 (w · ∇u)‖L2 ≤ C‖w · ∇u‖Ḣ1−α ≤ C ( ‖w‖Ḃ−α∞,∞‖∇u‖Ḣ1 + ‖∇u‖Ḃ−α∞,∞‖w‖Ḣ1 ) ≤ C ( ‖w‖Ḃ−α∞,∞‖∆u‖L2 + ‖w‖Ḃ−α∞,∞‖∇w‖L2 ) ≤ C‖w‖Ḃ−α∞,∞‖∇w‖L2 . EJDE-2021/89 LOGARITHMICALLY IMPROVED REGULARITY CRITERIA 7 Then we deduce, from (3.4) and the interpolation inequality, that 1 2 d dt ‖∇w(t)‖2L2 + ‖∆w(t)‖2L2 ≤ C‖w(t)‖Ḃ−α∞,∞‖∇w(t)‖L2‖(−∆) 1+α 2 (w)‖L2 ≤ C‖w(t)‖Ḃ−α∞,∞‖∇w(t)‖L2‖∇w(t)‖1−αL2 ‖∆w(t)‖αL2 ≤ C‖w(t)‖ 2 2−α Ḃ−α∞,∞ ‖∇w(t)‖2L2 + 1 2 ‖∆w(t)‖2L2 . Thus, d dt ‖∇w(t)‖2L2 ≤ C‖w(t)‖ 2 2−α Ḃ−α∞,∞ ‖∇w(t)‖2L2 = C ‖w(t)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(t)‖Ḣs0 ( 1 + log+ ‖w(t)‖Ḣs0 ) ‖∇w(t)‖2L2 . By Gronwall’s inequality, we obtain ‖∇w(t2)‖2L2 ≤ ‖∇w(t1)‖2L2 exp ( C log ( e+ sup t∈[t1,t2] ‖w(t)‖Ḣs0 ) × ∫ t2 t1 ‖w(τ)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(τ)‖Ḣs0 dτ ) (3.6) for all 0 < t1 < t2 < T . Moreover, it follows from (1.10) that for every ε > 0, there exists 0 < T ∗ < T such that∫ T T∗ ‖w(t)‖ 2 2−α Ḃ−α∞,∞ 1 + log+ ‖w(t)‖Ḣs0 dt < ε . This and (3.6) imply that there exists a constant C0 > 0 (independent of w(t)) such that ‖∇w(t)‖2L2 ≤ ‖∇w(T ∗)‖2L2 ( e+ sup t∈[T∗,t] ‖w(t)‖Ḣs0 )C0ε ≤ C ( e+ sup t∈[T∗,t] ‖w(t)‖Ḣs0 )C0ε , (3.7) for all T ∗ < t < T . To obtain the conclusion, it suffices to prove that ‖w(t)‖Ḣs0 is bounded on [T ∗, T ]. We can proceed as the proof of Theorem 1.1 of [5] and obtain 1 2 d dt ‖∆w(t)‖2L2 ≤ C (1 + ‖∇w(t)‖L2) 6 . (3.8) We now divide the rest of our proof into the two following cases: (i) If α ∈ (0, 1 2 ), then s0 ∈ (1, 3 2 ), and it follows from the inequality of Gagliardo- Nirenberg type that ‖w(t)‖Ḣs0 . ‖w(t)‖2−s0 Ḣ1 ‖w(t)‖s0−1 Ḣ2 . ‖∇w(t)‖2−s0L2 ‖∆w(t)‖s0−1 L2 . (3.9) Combining (3.7), (3.8), and (3.9) yields that there exists a constant C1 > 0 (inde- pendent of w(t)) such that ‖∇w(t)‖2L2 ≤ C + C sup τ∈[T∗,t] ‖∇w(τ)‖C1ε L2 , (3.10) 8 N. A. DAO, J. I. DÍAZ EJDE-2021/89 for all t ∈ [T ∗, T ). This implies that ‖∇w(t)‖2L2 is uniformly bounded in (T ∗, T ) if ε > 0 is chosen such that C1ε < 2. Therefore, w ∈ L∞(τ, T ;L6(R3)) , (3.11) for any τ ∈ (0, T ). Then, by the result by Beirão da Veiga [2], u is regular in (0, T ]. (ii) If α ∈ ( 1 2 , 1] then s0 ∈ ( 1 2 , 1]. Applying the inequality of Gagliardo–Nirenberg type we obtain ‖w(t)‖Ḣs0 . ‖w(t)‖1− s0 2 L2 ‖∆w(t)‖ s0 2 L2 . ‖∇u(t)‖1− s0 2 L2 ‖∆w(t)‖ s0 2 L2 . ( ‖u(t)‖1/2L2 ‖∆u(t)‖1/2L2 )1− s02 ‖∆w(t)‖ s0 2 L2 ≤ ( ‖u0‖1/2L2 ‖∆u(t)‖1/2L2 )1− s02 ‖∆w(t)‖ s0 2 L2 . ( ‖u0‖1/2L2 ‖∇w(t)‖1/2L2 )1− s02 ‖∆w(t)‖ s0 2 L2 . (3.12) Note that the last inequality was obtained by using the Biot-Savart law u(x, t) = C ∫ R3 K(x− y)w(y, t) dy , where K(x) is homogeneous of degree −2. As a result, ∇K(x) is a singular kernel of Calderón-Zygmund type. Note that ∆u(x, t) = C ∫ R3 ∇K(x− y) · ∇w(y, t) dy . It follows from the standard Calderón-Zygmund theory that ‖∆u(t)‖L2 . ‖∇w(t)‖L2 . A combination of (3.7), (3.8), and (3.12) implies that there exists a constant C2 > 0 such that ‖∇w(t)‖2L2 ≤ C + C sup τ∈[T∗,t] ‖∇w(τ)‖C2ε L2 , for t ∈ (T ∗, T ) . 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Nguyen Anh Dao Institute of Applied Mathematics, University of Economics Ho Chi Minh City, Ho Chi Minh City, Viet Nam Email address: anhdn@ueh.edu.vn Jesus Ildefonso D́ıaz Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain Email address: jidiaz@ucm.es 1. Introduction 2. Definitions and preliminary results 3. Proof of main results Acknowledgements References