Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation
Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation
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临界广义 KdV 方程的爆破曲线稳定性和爆破速率下限
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
F. Merle
中科院分区：
文献类型：
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作者：
Y. Martel;F. Merle
广义Korteweg-de弗里斯方程是由KdV方程导出的一类无穷维Hamilton系统，其中二次项被高阶幂项代替.这些方程在能量空间H1中有两个守恒律（L2范数和能量）.本文考虑临界广义KdV方程，它对应于非线性的最小幂，使得两个守恒律并不意味着所有H1解在时间上一致的H1界（从而全局存在）。根据[15]，对于这个方程确实存在解u（t），使得|u（吨）|H1 → +∞ as t ↑ T，其中T < +∞（我们称之为爆破解）。问题是要以定性的方式描述爆炸是如何发生的。对于L2质量接近于允许爆破的最小质量且在右边L2衰减的解，我们证明了在保持L2范数不变的情况下，经过重新标度和平移后，解在爆破时刻T局部收敛于空间中的一个普适轮廓.从这个轮廓的性质，我们改进了有限时间爆破解的爆破速率的标准下界。
The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)| H1 → +∞ as t ↑ T, where T < +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.