论文标题
关于粗糙非线性schrödinger方程的多泡爆破解决方案
On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations
论文作者
论文摘要
我们关注对重点质量质量质量案例中粗糙的非线性schrödinger方程的多泡爆炸解决方案。在一个和两个方面，我们构建了有限的时间多泡解决方案，该解决方案集中在$ k $不同的点，$ 1 \ leq k <\ k <\ infty $，并在伪造空间$σ$附近的伪造空间中的伪内形式爆破解决方案均匀地表现。渐近行为的上限与爆炸点处的噪声平坦度密切相关。此外，我们证明了在能量空间中的渐近行为$ h^1 $是$（t-t-t）^{3+ζ} $，$ζ> 0 $的情况下，在能量空间中的渐近行为$ h^1 $中的有条件唯一性。对于具有较低顺序扰动的非线性schrödinger方程，尤其是在没有经典的伪符合形式对称性和能量对话定律的情况下，也获得了这些结果。存在结果适用于规范确定性的非线性Schrödinger方程，并补充了先前的工作[43]。在随机情况和确定性情况下，条件唯一性结果都是新的。
We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate at $K$ distinct points, $1\leq K<\infty$, and behave asymptotically like a sum of pseudo-conformal blow-up solutions in the pseudo-conformal space $Σ$ near the blow-up time. The upper bound of the asymptotic behavior is closely related to the flatness of noise at blow-up points. Moreover, we prove the conditional uniqueness of multi-bubble solutions in the case where the asymptotic behavior in the energy space $H^1$ is of the order $(T-t)^{3+ζ}$, $ζ>0$. These results are also obtained for nonlinear Schrödinger equations with lower order perturbations, particularly, in the absence of the classical pseudo-conformal symmetry and the conversation law of energy. The existence results are applicable to the canonical deterministic nonlinear Schrödinger equation and complement the previous work [43]. The conditional uniqueness results are new in both the stochastic and deterministic case.
