Blowup of Smooth Solutions for an Aggregation Equation
We study the blowup criterion of smooth solutions for an inviscid aggregation equation in . By means of the losing estimates and the logarithmic Sobolev inequality, we establish an improved blowup criterion of smooth solutions.
In this paper, we consider the following aggregation equation in : with a given kernel . The unknown function 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 .
Laurent  has studied problem (1) in detail and proved several local and global existence results for a class of kernels 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 lead to the global in time existence of solutions (see e.g., [1, 4]). Nonsmooth kernels (and off the origin, like ) 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 . The authors of [6–10] studied the problem (1) with fractional dissipation and proved finite blowup of solutions or their global well-posedness for certain class of kernels. Recently, Karch and Suzuki  have classified kernels, which lead either to the blowup or global existence of solutions to (1) with the classical dissipation .
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  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 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 which contains the case . Here, we follow the ideas introduced in [2, 14–18]. Our main result reads as follows.
Theorem 1. Let , , , , . Suppose that is a smooth solution to (1). If there exists an absolute constant such that if then and the solution can be extended past time . In other words, if then the solution blows up at . Here, is a frequency localization on ; see Section 2.
Notation. Throughout the paper, stands for a generic constant. We will use the notation to denote the relation and the notation to denote the relations and .
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 be supported in the ring and such that We define also the function . Now for we set The following low-frequency cut-off will be also used:
Let us now recall the definition of Besov spaces through dyadic decomposition.
Let and ; then the inhomogeneous space is the set of tempered distributions such that To define the homogeneous Besov spaces we first denote by the space of tempered distributions modulo polynomials. Thus, we define the space as the set of distribution such that We point out that if , then we have and
In our next study, we require two kinds of coupled space-time Besov spaces. The first one is defined in the following manner: for and , we denote by the set of all tempered distributions satisfying The second mixed space is which is the set of tempered distribution satisfying We can define by the same way the spaces and .
The following embeddings are a direct consequence of Minkowski’s inequality.
Let , , and ; then we have
Now we give two useful lemmas.
Lemma 2 (Bernstein’s inequalities ). Let . Assume that ; then there exists a constant independent of , such that
Lemma 3 (logarithmic Sobolev inequality). Let , , and . Assume that . Then, the following inequality holds: where , for , , for , and is an absolute constant independent of .
The proof is rather standard and can be found in .
3. Proof of Theorem 1
Applying to (1), we have Multiplying (15) by and integrating the obtained equation in with respect to the space variable give Let us now turn to estimate the right-hand term of the previous equation by Bony’s decomposition . We decompose as a paraproduct For , integrating by parts together with the Hölder inequality yields We have similar estimates for using twice integration by parts: For , we have Adding (18)–(20), we infer that where we use the inequalities and . Thus, we deduce which implies that
Set Integrating (23) over with respect to time variable and then multiplying by the both obtained inequality, we get Let Taking the supremum over on both sides of inequality (25), we deduce that By the definition of , we know then we have Taking the sum over of (27) then using (29) and the Young inequality lead to Now if we choose large enough such that then Next we estimate . It is easy to obtain that Multiplying by , both sides of the inequality yields from which and (29) we have If , then Let us define This together with (32) and (36) implies that In particular, we have Applying Lemma 3 with and the embedding , we have For the sake of convenience, we denote noting that Plugging (40) into (39) then taking supremum over with respect to , we have We should point out that the previous inequality still holds if the time interval is replaced with . Thanks to the assumption (2) of Theorem 1, we deduce that Setting , we finally have If we choose , condition (2) ensures the term when , which implies that Hence, we have the regularity for the solution at and the solution can be continued after . This completes the proof of Theorem 1.
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).