Blowup of Smooth Solutions for an Aggregation Equation

Abstract

We study the blowup criterion of smooth solutions for an inviscid aggregation equation in ℝ^𝑛. By means of the losing estimates and the logarithmic Sobolev inequality, we establish an improved blowup criterion of smooth solutions.

1. Introduction

In this paper, we consider the following aggregation equation in ℝ^𝑛: 𝑢_𝑡(𝑢(𝑢()+∇⋅∇𝐾∗𝑢))=0,𝑥,0=𝑢₀(𝑥), (1) with a given kernel 𝐾∶ℝ^𝑛→ℝ. The unknown function 𝑢 is either the population density of a species or the density of particles in a granular medium. Aggregation equations of form (1) arise in many problems in biology, chemistry, and population dynamics and describe a collective motion and aggregation phenomena in biology and in mechanics of continuous media. From the mathematical point of view, (1) can be considered as a nonlinear, nonlocal transport equation, and its character depends strongly on properties of a given kernel 𝐾.

Laurent [1] has studied problem (1) in detail and proved several local and global existence results for a class of kernels 𝐾 with different regularity. Then, Bertozzi et al. [2–5] have proved finite-time blowup of solutions corresponding to compactly supported radial initial data. Those results can be summarized as follow. Kernels that are smooth (not singular) at origin 𝑥=0 lead to the global in time existence of solutions (see e.g., [1, 4]). Nonsmooth kernels (and 𝐶¹ off the origin, like 𝐾(𝑥)=𝑒^−|𝑥|) may lead to blowup of solutions either in finite or infinite time [1–4, 6, 7].

Equation (1) has been also intensively considered in the viscous case, namely, with the dissipative term (−Δ)^𝛾𝑢. The authors of [6–10] studied the problem (1) with fractional dissipation (−Δ)^𝛾/2𝑢 and proved finite blowup of solutions or their global well-posedness for certain class of kernels. Recently, Karch and Suzuki [11] have classified kernels, which lead either to the blowup or global existence of solutions to (1) with the classical dissipation Δ𝑢.

Typical approaches to prove a finite-time aggregation include an extension of the method of characteristics [4, 12], the energy method (e.g., [2, 3, 6, 7]) and the moment (or virial) method. The latter has been first applied to mean field models for self-gravitating particles and chemotaxis system [13] and recently in [8, 9, 11].

Our aim in this paper is to present another method showing finite time blowup of a large class of solutions of (1). In the mixed time-space Besov spaces, using the losing estimates and the logarithmic Sobolev inequality, we can set up the blowup criterion at some Δ_𝑗 which is the frequency localization operator in the Littlewood-Paley decomposition. The blowup result we obtained for (1) had been proved under the assumptions much relaxed compared to [6, 7, 12]. In addition, it allows us to consider potentials which are more general than those considered in previous papers, namely, we require ∇𝐾∈𝑊^1,1(ℝ^𝑛) which contains the case 𝐾=𝑒^−|𝑥|. Here, we follow the ideas introduced in [2, 14–18]. Our main result reads as follows.

Theorem 1. Let ∇𝐾∈𝑊^1,1(ℝ^𝑛), 𝑢₀∈𝐵^𝑠_𝑝,𝑞, 𝑠>𝑛/𝑝+1, 1<𝑝, 𝑞<∞. Suppose that 𝑢∈𝐶([0,𝑇);𝐵^𝑠_𝑝,𝑞)∩𝐶¹([0,𝑇);𝐵^𝑠−1_𝑝,𝑞) is a smooth solution to (1). If there exists an absolute constant 𝑀>0 such that if lim_𝜀→0sup_𝑗∈ℤ^𝑇_𝑇−𝜀Δ_𝑗𝑢_∞𝑑𝑡=𝛿<𝑀, (2) then 𝛿=0 and the solution 𝑢 can be extended past time 𝑡=𝑇. In other words, if lim_𝜀→0sup_𝑗∈ℤ^𝑇_𝑇−𝜀Δ_𝑗𝑢_∞𝑑𝑡≥𝑀, (3) then the solution blows up at 𝑡=𝑇. Here, Δ_𝑗 is a frequency localization on |𝜉|≈2^𝑗; see Section 2.

Note that 𝐵^𝑠−1_𝑝,𝑞 is a Banach algebra for 𝑠>𝑛/𝑝+1. One can easily prove that there exists a unique smooth solution 𝑢∈𝐶([0,𝑇);𝐵^𝑠_𝑝,𝑞)∩𝐶¹([0,𝑇);𝐵^𝑠−1_𝑝,𝑞) to (1) by standard method; see [19] for details.

Notation. Throughout the paper, 𝐶 stands for a generic constant. We will use the notation 𝐴≲𝐵 to denote the relation 𝐴≤𝐶𝐵 and the notation 𝐴≈𝐵 to denote the relations 𝐴≲𝐵 and 𝐵≲𝐴.

2. Preliminaries

In this preparatory section, we provide the definition of some function spaces based on the so-called Littlewood-Paley decomposition and we review some important lemmas that will be used constantly in the sequel.

We start with the dyadic decomposition. Let 𝜑∈𝐶^∞₀(ℝ^𝑛) be supported in the ring 𝒞∶={𝜉∈ℝ^𝑛,3/4≤|𝜉|≤8/3} and such that _𝑞∈ℤ𝜑2^−𝑞𝜉=1for𝜉≠0. (4) We define also the function 𝜒(𝜉)=1−∑_𝑞∈ℕ𝜑(2^−𝑞𝜉). Now for 𝑢∈𝒮^′ we set Δ₋₁𝑢=𝜒(𝐷)𝑢;∀𝑞∈ℕ,Δ_𝑞𝑢=𝜑2^−𝑞𝐷Δ̇𝑢,∀𝑞∈ℤ,_𝑞𝑢=𝜑2^−𝑞𝐷𝑢. (5) The following low-frequency cut-off will be also used: 𝑆_𝑞𝑢=_{−1≤𝑗≤𝑞−1}Δ_𝑗𝑆̇𝑢,_𝑞𝑢=_{𝑗≤𝑞−1}Δ̇_𝑗𝑢. (6)

Let us now recall the definition of Besov spaces through dyadic decomposition.

Let (𝑝,𝑞)∈[1,+∞]² and 𝑠∈ℝ; then the inhomogeneous space 𝐵^𝑠_𝑝,𝑞 is the set of tempered distributions 𝑢 such that ‖𝑢‖_𝐵^𝑠_𝑝,𝑞∶=2^𝑞𝑠Δ_𝑞𝑢_{𝐿𝑝ℓ𝑞}<∞. (7) To define the homogeneous Besov spaces we first denote by 𝒮^′/𝒫 the space of tempered distributions modulo polynomials. Thus, we define the space 𝐵̇^𝑠_𝑝,𝑟 as the set of distribution 𝑢∈𝒮^′/𝒫 such that ‖𝑢‖_̇𝐵^𝑠_𝑝,𝑞∶=2^𝑞𝑠Δ̇_𝑞𝑢_{𝐿𝑝ℓ𝑞}<∞. (8) We point out that if 𝑠>0, then we have 𝐵^𝑠_𝑝,𝑞=𝐵̇^𝑠_𝑝,𝑞∩𝐿^𝑝 and ‖𝑢‖_𝐵^𝑠_𝑝,𝑞≈‖𝑢‖_̇𝐵^𝑠_𝑝,𝑞+‖𝑢‖_𝐿^𝑝. (9)

In our next study, we require two kinds of coupled space-time Besov spaces. The first one is defined in the following manner: for 𝑇>0 and 𝑞≥1, we denote by 𝐿^𝑟_𝑇𝐵̇^𝑠_𝑝,𝑞 the set of all tempered distributions 𝑢 satisfying ‖𝑢‖_𝐿^𝑟_𝑇̇𝐵^𝑠_𝑝,𝑟∶=2^𝑞𝑠Δ̇_𝑞𝑢_{𝐿𝑝ℓ𝑞𝐿𝑟𝑇}<∞. (10) The second mixed space is 𝐿^𝑟_𝑇𝐵̇^𝑠_𝑝,𝑞 which is the set of tempered distribution 𝑢 satisfying ‖𝑢‖𝐿^𝑟_𝑇^̇_𝐵^𝑠_𝑝,𝑞∶=2^𝑞𝑠Δ̇_𝑞𝑢_{𝐿𝑟𝑇𝐿𝑝ℓ𝑞}<∞. (11) We can define by the same way the spaces 𝐿^𝑟_𝑇𝐵^𝑠_𝑝,𝑞 and 𝐿^𝑟_𝑇𝐵^𝑠_𝑝,𝑞.

The following embeddings are a direct consequence of Minkowski’s inequality.

Let 𝑠∈ℝ, 𝑟≥1, and (𝑝,𝑞)∈[1,∞]²; then we have 𝐿^𝑟_𝑇𝐵̇^𝑠_𝑝,𝑞↪𝐿^𝑟_𝑇𝐵̇^𝑠_𝑝,𝑞𝐿,if𝑞≥𝑟,_𝑟𝑇𝐵̇^𝑠_𝑝,𝑞↪𝐿^𝑟_𝑇𝐵̇^𝑠_𝑝,𝑞,if𝑟≥𝑞. (12)

Now we give two useful lemmas.

Lemma 2 (Bernstein’s inequalities [20]). Let 1≤𝑝≤𝑞≤∞. Assume that 𝑓∈𝐿^𝑝; then there exists a constant 𝐶 independent of 𝑓, 𝑗 such that supp𝑓⊂𝜉≤𝐶2^𝑗⟹𝜕^𝛼𝑓_𝐿𝑞≤𝐶2^{𝑗|𝛼|+𝑗𝑛((1/𝑝)−(1/𝑞))}𝑓_𝐿𝑝,1supp𝑓⊂𝐶2^𝑗≤𝜉≤𝐶2^𝑗⟹𝑓_𝐿𝑝≤𝐶2^{−𝑗|𝛼|}sup_𝛽=|𝛼|𝜕^𝛽𝑓_𝐿𝑝. (13)

Lemma 3 (logarithmic Sobolev inequality). Let 1≤𝑝<∞, 1≤𝑞<∞, and 𝑠>𝑛/𝑝+1. Assume that 𝑓∈𝐿¹_𝑇(𝐵̇⁰_∞,∞)∩𝐿^∞_𝑇(𝐵^𝑠−1_𝑝,𝑞). Then, the following inequality holds: ^𝑇₀𝑓(𝑡)_∞𝑑𝑡≤𝐶1+sup_𝑗^𝑇₀Δ_𝑗𝑓_∞𝑑𝑡1+log⁺𝑇𝑓_{𝐿∞𝑇(𝐵𝑠−1𝑝,𝑞)}, (14) where log⁺𝑥=log𝑥, for 𝑥>1, log⁺𝑥=0, for 𝑥≤1, and 𝐶 is an absolute constant independent of 𝑓,𝑇.

The proof is rather standard and can be found in [14].

3. Proof of Theorem 1

Applying Δ̇_𝑗 to (1), we have 𝜕_𝑡Δ̇_𝑗𝑢+∇⋅Δ̇_𝑗(𝑢(∇𝐾∗𝑢))=0. (15) Multiplying (15) by |Δ̇_𝑗𝑢|^𝑝−2Δ̇_𝑗𝑢 and integrating the obtained equation in ℝ^𝑛 with respect to the space variable give 1𝑝𝑑̇Δ𝑑𝑡^𝑗𝑢^𝑝_𝑝=−_ℝ𝑛∇⋅Δ̇_𝑗(𝑢(∇𝐾∗𝑢))Δ̇_𝑗𝑢^𝑝−2Δ̇_𝑗𝑢𝑑𝑥. (16) Let us now turn to estimate the right-hand term of the previous equation by Bony’s decomposition [21]. We decompose 𝑢(∇𝐾∗𝑢) as a paraproduct 𝑢(∇𝐾∗𝑢)=𝑇_𝑢∇𝐾∗𝑢+𝑇_{∇𝐾∗𝑢}𝑢+𝑅(𝑢,∇𝐾∗𝑢)=_𝑘𝑆̇_𝑘−1𝑢Δ̇_𝑘(∇𝐾∗𝑢)+_𝑘𝑆̇_𝑘−1(∇𝐾∗𝑢)Δ̇_𝑘𝑢+_{𝑘−𝑘′≤1}Δ̇_𝑘𝑢Δ̇_𝑘′(∇𝐾∗𝑢)=𝐼+𝐼𝐼+𝐼𝐼𝐼. (17) For 𝐼, integrating by parts together with the Hölder inequality yields ∇⋅Δ̇_ℝ𝑛𝑗(𝐼)Δ̇_𝑗𝑢^𝑝−2Δ̇_𝑗𝑢𝑑𝑥≲_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑢_𝐿∞Δ̇_𝑘(∇𝐾∗𝑢)_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−2_𝑝∇Δ̇_𝑗𝑢_𝐿𝑝≲_{𝑘−𝑗≤4}2^𝑗−𝑘𝑆̇_𝑘−1𝑢_𝐿∞Δ̇_𝑘∇²𝐾∗𝑢_𝐿𝑝×Δ̇_𝑗𝑢_𝐿^𝑝−2_𝑝Δ̇_𝑗𝑢_𝐿𝑝≲_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑢_𝐿∞Δ̇_𝑘𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−1_𝑝. (18) We have similar estimates for 𝐼𝐼 using twice integration by parts: ∇⋅Δ̇_ℝ𝑛𝑗(𝐼𝐼)Δ̇_𝑗𝑢^𝑝−2Δ̇_𝑗𝑢𝑑𝑥≲_{𝑘−𝑗≤4ℝ𝑛}𝑆̇_𝑘−1(∇𝐾∗𝑢)Δ̇_𝑘𝑢Δ̇_𝑗𝑢^𝑝−2∇Δ̇_𝑗≲𝑢𝑑𝑥_{𝑘−𝑗≤4ℝ𝑛}𝑆̇_𝑘−1(∇𝐾∗𝑢)∇Δ̇_𝑗𝑢^𝑝𝑑𝑥≲_{𝑘−𝑗≤4ℝ𝑛}𝑆̇_𝑘−1(Δ𝐾∗𝑢)Δ̇_𝑗𝑢^𝑝≲𝑑𝑥_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑢_𝐿∞Δ̇_𝑗𝑢^𝑝_𝐿𝑝. (19) For 𝐼𝐼𝐼, we have ∇⋅Δ̇_ℝ𝑛𝑗(𝐼𝐼𝐼)Δ̇_𝑗𝑢^𝑝−2Δ̇_𝑗𝑢𝑑𝑥≲_{𝑘−𝑘′≤1𝑘≥𝑗−3}Δ̇_𝑘𝑢_𝐿∞Δ̇_𝑘′(∇𝐾∗𝑢)_𝐿𝑝×Δ̇_𝑗𝑢_𝐿^𝑝−2_𝑝∇Δ̇_𝑗𝑢_𝐿𝑝≲_{𝑘−𝑘′≤1𝑘≥𝑗−3}2^𝑗−𝑘Δ̇_𝑘𝑢_𝐿∞Δ̇_𝑘′∇²𝐾∗𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−1_𝑝≲_{𝑘≥𝑗−3}Δ̇_𝑘𝑢_𝐿∞Δ̇_𝑘𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−1_𝑝. (20) Adding (18)–(20), we infer that 1𝑝𝑑̇Δ𝑑𝑡^𝑗𝑢^𝑝_𝐿𝑝≲_𝑗∈ℤ𝑆̇_𝑘−1𝑢_𝐿∞Δ̇_𝑗𝑢^𝑝_𝐿𝑝+_{𝑘≥𝑗−3}Δ̇_𝑘𝑢_𝐿∞Δ̇_𝑘𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−1_𝑝≲_𝑗∈ℤ‖𝑢‖_𝐿^∞Δ̇_𝑗𝑢^𝑝_𝐿𝑝+_{𝑘≥𝑗−3}‖𝑢‖_𝐿^∞Δ̇_𝑘𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑝−1_𝑝, (21) where we use the inequalities ‖𝑆̇_𝑘−1𝑢‖_𝐿∞≤‖𝑢‖_𝐿^∞ and ‖Δ̇_𝑘𝑢‖_𝐿∞≤‖𝑢‖_𝐿^∞. Thus, we deduce 𝑑̇Δ𝑑𝑡^𝑗𝑢_𝑝≲_𝑗∈ℤ‖𝑢‖_𝐿^∞Δ̇_𝑗𝑢_𝐿𝑝+_{𝑘≥𝑗−3}‖𝑢‖_𝐿^∞Δ̇_𝑘𝑢_𝐿𝑝, (22) which implies that 𝑑̇Δ𝑑𝑡^𝑗𝑢^𝑞_𝐿𝑝≲𝑞_𝑗∈ℤ‖𝑢‖_𝐿^∞Δ̇_𝑗𝑢^𝑞_𝐿𝑝+𝑞_{𝑘≥𝑗−3}‖𝑢‖_𝐿^∞Δ̇_𝑘𝑢_𝐿𝑝Δ̇_𝑗𝑢_𝐿^𝑞−1_𝑝. (23)

Set Ψ_𝜆𝑡^′,𝑡=𝜆^𝑡′_𝑡𝑢𝑡^′′_∞𝑑𝑡^′′,Ψ_𝜆(𝑡)=Ψ_𝜆(0,𝑡). (24) Integrating (23) over [0,𝑡) with respect to time variable 𝜏 and then multiplying by 2^{𝑞(𝑗𝑠−Ψ𝜆(𝑡))} the both obtained inequality, we get 2^{𝑞(𝑗𝑠−Ψ𝜆(𝑡))}Δ̇_𝑗𝑢^𝑞_𝐿𝑝(𝑡)≲2^𝑗𝑞𝑠Δ̇_𝑗𝑢^𝑞_𝐿𝑝(0)+^𝑡₀2^{−𝑞Ψ𝜆(𝜏,𝑡)}‖𝑢‖_𝐿^∞(𝜏)×_𝑗∈ℤ2^{𝑞(𝑗𝑠−Ψ𝜆(𝜏))}Δ_𝑗𝑢^𝑞_𝐿𝑝+_{𝑘≥𝑗−3}2^{(𝑗−𝑘)𝑠}2^{(𝑞−1)(𝑗𝑠−Ψ𝜆(𝜏))}×Δ̇_𝑗𝑢_𝐿^𝑞−1_𝑝2^{𝑘𝑠−Ψ𝜆(𝜏)}Δ̇_𝑘𝑢_𝐿𝑝𝑑𝜏. (25) Let 𝛼_𝑗,𝑇=sup_{𝑡∈[0,𝑇)}2^{𝑗𝑠−Ψ𝜆(𝑡)}Δ̇_𝑗𝑢_𝑝,𝛼_𝑇=𝛼_{𝑗,𝑇𝑙𝑞}. (26) Taking the supremum over [0,𝑇) on both sides of inequality (25), we deduce that 𝛼^𝑞_𝑗,𝑇≲𝛼^𝑞_𝑗,0+sup_{[)𝑡∈0,𝑇}^𝑡₀2^{−𝑞Ψ𝜆(𝜏,𝑡)}‖𝑢‖_𝐿^∞×𝛼𝑑𝜏_{𝑞−1𝑗,𝑇𝑗∈ℤ}𝛼_𝑘,𝑇+𝛼^𝑞−1_{𝑗,𝑇𝑘≥𝑗−3}2^{(𝑗−𝑘)𝑠}𝛼_𝑘,𝑇. (27) By the definition of Ψ_𝜆(𝜏,𝑡), we know 𝑑2𝑑𝜏^{−𝑞Ψ𝜆(𝜏,𝑡)}=𝑞𝜆log2⋅2^{−𝑞Ψ𝜆(𝜏,𝑡)}‖𝑢‖_∞(𝜏); (28) then we have ^𝑡₀2^{−𝑞Ψ𝜆(𝜏,𝑡)}‖𝑢‖_∞𝑑𝜏=^𝑡₀1𝑑𝑞𝜆log22𝑑𝜏^{−𝑞Ψ𝜆(𝜏,𝑡)}=1𝑞𝜆log21−2^{−𝑞Ψ𝜆(0,𝑡)}≤1.𝑞𝜆log2 (29) Taking the sum over 𝑗 of (27) then using (29) and the Young inequality lead to 𝛼^𝑞_𝑇≲𝑢₀^𝑞_{̇𝐵𝑠𝑝,𝑞}+1𝛼𝑞𝜆log2^𝑞𝑇. (30) Now if we choose 𝜆 large enough such that 2𝜆>,𝑞log2 (31) then 𝛼_𝑇≲𝑢₀^𝑞_{̇𝐵𝑠𝑝,𝑞}. (32) Next we estimate ‖𝑢‖_𝑝. It is easy to obtain that ‖𝑢(𝑡)‖_𝐿^𝑝≲𝑢_0𝐿𝑝+^𝑡₀‖𝑢(𝜏)‖_𝐿^∞‖𝑢(𝜏)‖_𝐿^𝑝𝑑𝜏. (33) Multiplying by 2^{−Ψ𝜆(𝑡)}, both sides of the inequality yields 2^{−Ψ𝜆(𝑡)}‖𝑢(𝑡)‖_𝐿^𝑝≲𝑢_0𝐿𝑝+^𝑡₀2^{−Ψ𝜆(𝜏,𝑡)}‖𝑢(𝜏)‖_𝐿^∞2^{−Ψ𝜆(𝜏)}‖𝑢(𝜏)‖_𝐿^𝑝𝑑𝜏, (34) from which and (29) we have sup_{[)𝑡∈0,𝑇}2^{−Ψ𝜆(𝑡)}‖𝑢(𝑡)‖_𝐿^𝑝≲𝑢_0𝐿𝑝+1𝑞𝜆log2sup_{[)𝑡∈0,𝑇}2^{−Ψ𝜆(𝑡)}‖𝑢(𝑡)‖_𝐿^𝑝. (35) If 𝜆≳2/log2, then sup_{𝑡∈[0,𝑇)}2^{−Ψ𝜆(𝑡)}‖𝑢(𝑡)‖_𝑝≲𝑢_0𝑝. (36) Let us define 𝛽_𝑇=sup𝛼_𝑇,sup_{[)𝑡∈0,𝑇}2^{−Ψ𝜆(𝑡)}‖𝑢(𝑡)‖_𝑝. (37) This together with (32) and (36) implies that 𝛽_𝑇≲𝑢_{0𝐵𝑠𝑝,𝑞}. (38) In particular, we have ‖𝑢(𝑡)‖_𝐵^𝑠_𝑝,𝑞≲2^Ψ𝜆(𝑡)𝑢_{0𝐵𝑠𝑝,𝑞},∀𝑡∈[0,𝑇). (39) Applying Lemma 3 with 𝑓(𝑡)=𝑢(𝑡) and the embedding 𝐵^𝑠_𝑝,𝑞↪𝐵^𝑠−1_𝑝,𝑞, we have ^𝑡₀‖𝑢‖_𝐿^∞𝑑𝜏≲1+sup_𝑗^𝑡₀Δ_𝑗𝑢_𝐿∞×𝑑𝜏1+log⁺𝑡‖𝑢‖_𝐿^∞_𝑡(𝐵^𝑠_𝑝,𝑞). (40) For the sake of convenience, we denote 𝜁(𝑇)=sup_[)0,𝑇‖𝑢(𝑡)‖_𝐵^𝑠_𝑝,𝑞, (41) noting that Ψ_𝜆(𝑡)=𝜆^𝑡₀‖𝑢‖_∞𝑑𝜏. (42) Plugging (40) into (39) then taking supremum over [0,𝑇) with respect to 𝑡, we have 𝜁(𝑇)≲2^{𝜆(1+sup𝑗∫𝑇0‖Δ𝑗𝑢‖𝐿∞𝑑𝜏(1+log+(𝑇𝜁(𝑇))))}𝜁(0). (43) We should point out that the previous inequality still holds if the time interval [0,𝑇) is replaced with [𝑇−𝜀,𝑇). Thanks to the assumption (2) of Theorem 1, we deduce that 𝜁(𝑇)≲2^{𝜆sup𝑗∫𝑇𝑇−𝜀‖Δ𝑗𝑢‖∞𝑑𝜏log+(𝜀𝜁(𝑇))}𝜁(𝑇−𝜀). (44) Setting 𝑍(𝑇)=log(𝑒+𝜁(𝑇)), we finally have 𝑍(𝑇)≲𝜆sup_𝑗^𝑇_𝑇−𝜀Δ_𝑗∇𝑢_∞𝑑𝜏𝑍(𝑇)+𝑍(𝑇−𝜀). (45) If we choose 𝑀=1/𝐶𝐶₀𝜆, condition (2) ensures the term 𝜆sup_𝑗∫^𝑇_𝑇−𝜀‖Δ_𝑗𝑢‖_∞𝑑𝜏<1/𝐶  when 𝜀→0, which implies that 𝑍(𝑇)≲𝑍(𝑇−𝜀). (46) Hence, we have the 𝐵^𝑠_𝑝,𝑞 regularity for the solution at 𝑡=𝑇 and the solution can be continued after 𝑡=𝑇. This completes the proof of Theorem 1.

Acknowledgments

This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant no. 50925727, the National Defense Advanced Research Project Grants nos. C1120110004, 9140A27020211DZ5102 the Key Grant Project of Chinese Ministry of Education under Grant no. 313018, and the Fundamental Research Funds for the Central Universities (2012HGCX0003).