Changxing Miao, Guixiang Xu, Lifeng Zhao
We consider the defocusing energy-critical nonlinear Schrödinger equation of fourth order $iu_t+Δ^2 u=-|u|^\frac{8}{d-4}u$. We prove that any finite energy solution is global and scatters both forward and backward in time in dimensions $d\geq9$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that the renormalized defocusing mKdV equation on the circle is locally in time $C^{0}$-wellposed on the Fourier Lebesgue space ${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytic…
We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy subcritical defocusing, and $L^2$ subcritical (de)focusing GP hierarchies, in spaces also used by Erdös, Schlein and Yau in \cite{esy1,esy2}. By use of this tool, we prove a priori $H^1$ bounds for positive semidefinite solutions in those spaces. Moreover, we obtain global well-posedness results for positive semidefinite solutions in the spaces studied in the wo…
Almost sharp global wellposedness and scattering for the defocusing conformal wave equation on the hyperbolic space
该研究利用双曲几何增强的 Morawetz 与 Strichartz 估计，证明了 $d\geq3$ 维双曲空间中散焦共形非线性波动方程在低正则径向初值下的近乎尖锐全局适定性与散射。
We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order $iu_t+Δ^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|Δu(t)\|_{2}<\|ΔW\|_{2}$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.
We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i\partial_t u=-\epsΔu+Δ^2 u+P((\partial_x^αu)_{\absα\ls 2}, (\partial_x^α\bar{u})_{\absα\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate with a smoothing effect on exterior domains.
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schrödinger equation with harmonic potential, $(i\partial_t + \frac Δ2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free solution in the space $Σ=H^1\bigcap\mathcal F H^1$. We preclude the concentration of energy in finite time by combining the energy decay estimates.
In this paper, we show the local well-posedness of the generalized Boussinesq equation(gBQ) in $L^{2}(\mathbb{R}^d), H^{1}(\mathbb{R}^d)$ and obtain the global well-posedness, finite-time blowup and small initial data scattering of gBQ in energy space $H^1(\mathbb{R}^d)$. Moreover, we obtain the large radial initial data scattering of defocusing case for $ d\geq 3 $ by using the method of Dodson-Murphy.
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
In this article we consider the initial value problem for the m-equivariant Chern-Simons-Schrödinger model in two spatial dimensions with real-valued coupling parameter g. This is a covariant NLS type problem that is L^2-critical. We prove that at the critical regularity, for any integer-valued equivariance index m, the initial value problem in the defocusing case (g < 1) is globally wellposed and the solution scatters. The problem is focusing when g >= 1, and in this case we prove that for nonnegative integer-valued equivariance indices m there exist constants c = c_{m, g} such that, at the c…
We study the cubic defocusing nonlinear Schrödinger equation on $\mathbb{R}^4$ with supercritical initial data. For randomized initial data in $H^s(\mathbb{R}^4)$, we prove almost sure local wellposedness for $\frac{1}{7} < s < 1$ and almost sure scattering for $\frac{5}{7} < s < 1$. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and - for the almost sure scattering result - an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, w…
In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schrödinger equation. Using the $I$-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in $H^γ(\mathbb{R}^d), 5\leq d\leq 7$ with $γ(d)<γ<2$, where $γ(5)=\frac{8}{5}, γ(6)=\frac{5}{3}$ and $γ(7)=\frac{13}{7}$.
In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{ 1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compact…
We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrodinger equation in high dimensions n>4, for general L^2 initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the Ground State in the focusing case.
Wellposedness and asymptotic behavior of solutions for the quintic wave equation with nonlocal dissipation
本文研究了带有能量临界非线性项和非局部阻尼机制的半线性波方程，通过构造弱解并利用能量估计和紧致性论证，探讨了非局部阻尼对解的稳定性和衰减行为的影响。
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearit…
In this paper we study the local and global regularity properties of the cubic nonlinear Schrödinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS \cite{et2} and the cubic NLS on the line \cite{erin}. We also prove …
In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) $iu_t+Δ^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if $p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schrödinger e…
We consider the energy critical nonlinear Schrödinger equation in dimensions $d \ge 3$ with a harmonic oscillator potential $V(x) = \tfrac{1}{2} |x|^2$. When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space $Σ$, consisting of all functions $u_0$ such that both $\nabla u_0$ and $x u_0$ belong to $L^2$. This result extends a theorem of Killip-Visan-Zhang \cite{kvz_quadratic_potentials}, which treats the radial case. For the focusing problem, we obtain global wellposedness for all data satisfying an analogue of the usual size restriction in te…
We establish an interaction Morawetz estimate for the magnetic Schrödinger equation for $n\geq 3$ under certain smallness conditions on the gauge potentials, but with almost optimal decay. As an application, we prove global wellposedness and scattering in $H^1$ for the cubic defocusing magnetic Schrödinger equation for $n = 3.$
We prove that the solution map, associated to the quintic fourth order nonlinear Schrödinger equation, exhibits the norm inflation phenomenon at every point in the Sobolev spaces of super-critical regularity. Indeed, we prove this result separately in the cases of negative and of positive regularity. In the negative regularity case, we prove the result for both the defocusing and focusing equations: in the one dimensional case, the associated solution map exhibits norm inflation in Sobolev spaces of super-critical regularity (including the critical index); in the higher dimensional case, the s…
We investigate the cubic defocusing fourth order Schrödinger equation $iu_t + Δ^2u + |u|^2u=0$ in arbitrary space dimension $\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n \le 8$ and ill-posed when $n \ge 9$, with the additional important information that scattering holds true when $5 \le n \le 8$.
We prove large-data scattering and existence of wave operators in the energy space for the systems of $N$ defocusing fourth-order Schrödinger equations with mass-supercritical and energy-subcritical power-type nonlinearity. In addition, new nonlinear interaction Morawetz identities and inequalities are given, suitable to shed lights on the decay of the solution with respect some Lebesgue norms when the space dimensions are $d=3,4$.
We prove the decay in the energy space for the solution to the defocusing biharmonic Hartree-Fock equations with mass-supercritical and energy-subcritical Choquard-type nonlinearity in space dimension $d\geq3$. We treat both the free and the perturbed by a potential case. As a direct consequence, we obtain large-data scattering in $H^2(\mathbb R^d)^N, \, N\geq 1$.
Global Well-posedness for The Fourth-order Nonlinear Schrödinger Equations on $\mathbb{R}^{2}$
本文利用$I$-方法证明了二维散焦四阶非线性Schrödinger方程在$H^{s}(\mathbb{R}^{2})$中当$2-\frac{3}{4k}<s<2$时的全局适定性。
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case
针对径向数据，研究组合非线性薛定谔方程在阈值以下的全局适定性与散射，利用轮廓分解和集中紧性方法，在$d\leq4$时建立结果。
Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case
本文研究了带有势能的第四阶非线性薛定谔方程在n≥5维空间中的全局良态性，证明了径向初值在H²空间下的存在性和散射性，采用Strichartz估计、Sobolev范数等方法扩展了无势能情况下的结果。
In this paper, we prove the global wellposedness of the Navier-Stokes equations describing a motion of compressible, viscous, barotropic fluid flow in a 3 dim. exterior domain in the $L_p$ in time and $L_2 \cap L_6$ maximal regularity framework. This is an extension of a famous thoerem due to Matsumura-Nishida Commun Math. Phys. 89 (1983), 445--464. In Matsumura and Nishida theory, they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to $L_2$ in space-time. On the ot…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。