FINITE-TIME SINGULARITIES OF AN AGGREGATION EQUATION IN R WITH FRACTIONAL DISSIPATION
FINITE-TIME SINGULARITIES OF AN AGGREGATION EQUATION IN R WITH FRACTIONAL DISSIPATION
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发表时间：
2008
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通讯作者：
Li Dong;José L. Rodrigo
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作者：
Li Dong;José L. Rodrigo
我们考虑在n≥2的Rn中具有分数阶耗散的聚集方程，即ut +∇·(u∇K∗u) =−ν(−∆)γ/2u，其中0≤γ≤2，K是一个非负递减的径向核，在原点有一个Lipschitz点，例如K(x) = e−|x|。证明了对于一般初始数据，当0≤γ < 1时，解在有限范围内爆破。相反，我们证明了当1 < γ≤2时，方程是全局适定的。
We consider an aggregation equation in Rn, n ≥ 2, with fractional dissipation, namely, ut + ∇ · (u∇K ∗ u) = −ν(−∆)γ/2u , where 0 ≤ γ ≤ 2 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove that for 0 ≤ γ < 1 the solutions develop blow-up in finite for a general class of initial data. In contrast we prove that for 1 < γ ≤ 2 the equation is globally well-posed.