论文标题
$ l^2 $ - 关键和$ l^2 $ -Superital Perfermitical Permerized Hartree方程中的稳定爆炸动力学
Stable blow-up dynamics in the $L^2$-critical and $L^2$-supercritical generalized Hartree equation
论文作者
论文摘要
我们研究了具有径向对称性的广义Hartree方程中的稳定爆炸动力学，具有非局部，卷积类型的非线性的schrödinger-type方程：$ iu_t +ΔU + +\ lest（| x | x | x |^|^{ - （d-2）}} \ ast} \ ast | ust | u | u | u | u | \ mathbb {r}^d $。首先，我们考虑$ l^2 $ - 批判性案例d = 3 = 3、4、5、6、7，并获得一般的爆破具有自相似结构，不仅表现出平方根的爆炸率，还表现出对数字校正的校正（通过渐近分析和功能拟合）。在这种情况下，我们还研究了爆炸配置文件，并表明通用爆破解决方案会融合到重新固定的$ q $，这是椭圆方程的基态解决方案$-ΔQ+q-+q- \ left（| x |^{ - （d-2）} \ ast | q | q |^p |^p \ p \ right） 我们还考虑了$ l^2 $ - 在尺寸d = 3,4中的秘密情况。我们得出了自相似爆炸的轮廓方程，并确定其解决方案的存在和局部唯一性。就像在NLS $ l^2 $ - 超临界方面一样，配置文件方程式展示了非振荡，多项式衰减（多重凹跃）溶液的分支。在查找多重倾斜溶液的过程中，应用了将约束置于解决相应ode的数值方案。通过动态缩放方法对通用Hartree方程的解决方案的直接数值模拟表明$ q_ {1,0} $是稳定爆炸的配置文件。在这种超临界情况下，我们在没有任何校正的情况下获得了爆炸率。这种爆炸发生在聚焦级别$ 10^{-5} $上，因此在数值上可观察到（与$ l^2 $ - 临界情况不同）。 总而言之，我们发现结果类似于相应的NLS设置中稳定爆炸动力学的行为。因此，人们可能会期望Schrödinger-type方程中的非线性形式对于稳定的爆炸不是必不可少的。
We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+Δu +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the $L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate but also the log-log correction (via asymptotic analysis and functional fitting). In this setting we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled $Q$, a ground state solution of the elliptic equation $-ΔQ+Q- \left(|x|^{-(d-2)} \ast |Q|^p \right) |Q|^{p-2} Q =0$. We also consider the $L^2$-supercritical case in dimensions d=3,4. We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS $L^2$-supercritical regime, the profile equation exhibits branches of non-oscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ODE is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the $Q_{1,0}$ is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level $10^{-5}$, and thus, numerically observable (unlike the $L^2$-critical case). In summary, we find that the results are similar to the behavior of stable blowup dynamics in the corresponding NLS settings. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in stable blow-up.
