Xavier Carvajal
We prove that, the initial value problem associated to u_{t} + i\alphau_{xx} + βu_{xxx} + iγ|u|^{2}u = 0, x,t \in R, is locally well-posed in Sobolev spaces H^{s} for s>-1/4.
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We study the local well-posedness of the initial-value problem for the nonlinear "good" Boussinesq equation with data in Sobolev spaces \textit{$H^s$} for negative indices of $s$.
In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves Saut-Tzvetkov's one and our global well-posedness gives an affirmative answer to Saut-Tzvetkov's $L^2$-data conjecture.
In this paper we study the existence of solutions and their concentration phenomena of a singularly perturbed semilinear Schrodinger equation with the presence of the critical Sobolev exponent.
In this paper we obtain a stabilization result for the Schrödinger equation under generic assumptions on the potential. Then we consider the Schrödinger equation with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of system is almost surely non-bounded in Sobolev spaces.
We consider a system of coupled cubic Schrödinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schrödinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times.
In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.
In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schrödinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof of the linear profile decomposition for the Schröindger equation with initial data in the homogeneous Sobolev space; as a consequence, there exists a maximizer for the Sobolev-Strichartz inequality.
In this paper we consider the cubic Schrodinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in H^s for s>131/416. We also obtain improved growth bounds for higher order Sobolev norms.
We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some non-admissible retarded estimates, a Hörmander type spectral multiplier theorem, and Keller type eigenvalue bounds with complex-valued potentials are also obtained.
We study the energy cascade problematic for some nonlinear Schrödinger equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one $|u|^2u$. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the f…
In this paper, we consider in $R^n$ the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\frac{2n}{n+1}<p<2$ and $s>n(1-\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.
Normalized solutions of quasilinear Schrödinger equation with Sobolev critical exponent on star-shaped bounded domains
本文研究星形有界域上具Sobolev临界指数的拟线性Schrödinger方程，通过双偶方法证明了基态与山路解两类正归一化解的存在性。
It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.
We consider the cubic nonlinear Schrödinger equation on product manifolds $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions. The main ingredient of the proof is to establish an iteration bound, which is based on the idea used by Bourgain in \cite{B1}.
We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}
The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct an almost conserved quantity using multilinear correction term following the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle the resonant interactions which are significant due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear…
Let $σ\in(0,1)$ with $σ\neq\frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: $$i\partial_tu+(-Δ)^σu+μ|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-Δ)^σ$ is the Fourier multiplier of symbol $|ξ|^{2σ}$, and $μ=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast …
Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$ norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in $H^s$.
Normalized solutions for a class of Sobolev critical Schrodinger systems
本文研究一类具有Sobolev临界耦合项的耦合Schrödinger系统，在明确条件下建立了归一化解的存在性与多重性结果，并给出散焦情形的非存在性结果，为Sobolev临界系统归一化解提供了更全面的理解。
Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential
本文研究了二维立方非线性薛定谔方程在部分谐振子势下的Sobolev范数增长，通过双线性估计和特定方法证明了解的多项式界。
We study the $L^p$-theory for the Schrödinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with those defined via $(-Δ)^{s/2}$. We consider all regularities $0<s<2$. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator $\mathcal L_a$.
Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schrödinger Equations
本文证明了加权分数阶Sobolev空间的紧嵌入，并以此作为应用，通过变分方法证明了在无穷远处函数可消失的非线性Schrödinger方程解的存在性。
In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}+V(x)m(u)u=f(x,u),\ x\in\mathbb{R}^{N},$$ where $s\in ]0,1[$, $N\geq2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u| \…
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrödinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2σq^2(x,t)\bar q(-x,t)=0,\quadσ=\pm1,\\ q(x,0)&=q_0(x). \end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $\bar\partial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.
In this paper and the companion work \cite{LIZE2}, we prove that the Schrödinger map flows from $\Bbb R^d$ with $d\ge 2$ to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper with the companion work \cite{LIZE2} solves the open problem …
In this paper we prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth \begin{eqnarray*} -Δu-λc(x)u-κα(Δ(|u|^{2α}))|u|^{2α-2}u = |u|^{q-2}u+|u|^{2^*-2}u,\quad u\in{D^{1,2}(\mathbb{R}}^N), \end{eqnarray*} via variational methods, where $λ\geq0$, $0<α<1/2$, $2<q<2^*$. It is interesting that we do not need to add a weight function to control $|u|^{q-2}u$.
We study the spectrum of Schrödinger operators with matrix valued potentials utilizing tools from infinite dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the m…
Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on $\R^2$ and $\R^3$
利用倒置I-方法和更高阶修正能量方法，证明了二维和三维空间中具有三次和Hartree型非线性项的散焦分数阶非线性薛定谔方程解的高Sobolev范数的多项式增长界。
We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equati…
This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} Δ^{2}u-λu=μ|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $μ>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $λ$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are p…
In this paper we consider linear, time dependent Schrödinger equations of the form $i \partial_t ψ= K_0 ψ+ V(t) ψ$, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations $V(t)$ such that the Hamiltonian $K_0 + V(t)$ generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on $\mathbb R$, (ii) the half-wave equation on $\mathbb T$ and (iii) the Dirac-Schrödinger equation on the sphere. In each case, $V(t)$ is a smooth and p…
In this paper we deal with the nonlinear Schrödinger system \[ -Δu_i =μ_i u_i^3 + βu_i \sum_{j\neq i} u_j^2 + λ_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(Ω) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($β<0$ fixed or $β\to -\infty$) or in the weakly cooperative case ($β\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $Ω$, families of positive solutions which blowup and concentrate at different points as $λ_1,\ldots, λ_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenbe…
Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
本文研究ℝ^N中Sobolev临界情形下零质量拟线性Schrödinger与高阶分数阶Poisson强耦合系统，借助变号对数Riesz核的Choquard方程及多项式核一致逼近的变分近似方法证明正解存在。
This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.
We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrödinger equation with mixed power nonlinearities \begin{equation*} i \partial_t v + Δv + μv |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \end{equation*} where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $μ> 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, …
We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schrödinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.
Normalized solutions on large smooth domains to the Schrödinger equations with potential and combined nonlinearities: The Sobolev critical case
本文研究含位势和混合非线性项的薛定谔方程在 Sobolev 临界情形下指定质量解的存在性与多重性，克服了 Pohozaev 恒等式方法因位势项存在而失效的困难，补充了 Bartsch-Qi-Zou 的相关工作。
We prove an abstract theorem giving a $\langle t\rangle^ε$ bound ($\forall ε>0$) on the growth of the Sobolev norms in linear Schrödinger equations of the form $i \dot ψ= H_0 ψ+ V(t) ψ$ when the time $t \to \infty$. The abstract theorem is applied to several cases, including the cases where (i) $H_0$ is the Laplace operator on a Zoll manifold and $V(t)$ a pseudodifferential operator of order smaller then 2; (ii) $H_0$ is the (resonant or nonresonant) Harmonic oscillator in $R^d$ and $V(t)$ a pseudodifferential operator of order smaller then $H_0$ depending in a quasiperiodic way on time. The p…
We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}Δu + V(x)u + ψu = λ|u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}Δψ= {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. \] where $\varepsilon > 0$ is a small positive parameter, $λ> 0$, $3 < p \le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\varepsilon} \in {H^1}({\mathbb{R}^3})$ which concentrates around a loca…
We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local $L^2$-mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scali…
Traveling waves for nonlinear Schrödinger equations
该研究针对亚声速非线性薛定谔方程的行波解问题，采用含动量的Sobolev型不等式变体方法，证明了对应方程极小值子的存在性，覆盖多种含Sobolev次临界非线性效应的物理模型。
On the embedding of weighted Sobolev spaces with applications to a planar nonlinear Schrödinger equation
本文研究带发散权的加权Sobolev空间的嵌入性质，重点分析权相等时的维数依赖特性及二维临界情形，并将结果应用于带强制势的平面非线性Schrödinger方程解的存在性证明。
We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schrödinger equation with mixed power nonlinearities $$ i \partial_t ϕ+ Δϕ+ μϕ|ϕ|^{q-2} + ϕ|ϕ|^{2^* - 2} = 0, \quad (t, x) \in R \times R^N, $$ where $N \geq 3$, $ϕ: R \times R^N \to C$, $μ> 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. It was already proved that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is…
Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
本文研究二维分数阶 Sobolev 极限情形下 Schrödinger-Poisson 系统的解的存在性与定性性质，通过辅助 Choquard 方程的变分逼近方法证明解存在，并利用移动平面技巧得到对称性与衰减结果。
We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on $\mathbb{R}^d$. These results extend the previous ones in \cite{HongSire} for $σ\geq 2$. This covers the well-known result for the Schrödinger equation $σ= 2$ given in \cite{CazenaveWeissler}. In the case $σ\in (0,2)\backslash \{1\}$, we give the local well-posedness in sub-critical case for all exponent $ν> 1$ in contrast of ones in \cite{HongSire}. This also generalizes the ones of \cite{ChoHwangKwonLee} when $d = 1$ and of \cite{GuoHuo} when $d \geq 2$ whe…
We consider the following wave guide nonlinear Schrödinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T} \_y$. We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. The proof is an adaptation of the method of Hani--Pausader--Tzvetkov--Visciglia. Combining this scattering theory with a recent result by Gérard--Grellier, we infer existence of global solutions to (WS) which are unbounded in t…
Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential
本文研究了Sobolev临界Schrödinger方程在捕获势下的归一化解的存在性和多重性，包括局部最小解和山峰解的构造。
Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces
针对一维和二维环面上带二次非线性 $\overline{u}^2$ 的 NLS，本文通过修正 $X^{s,b}$ 空间，在常规双线性估计失效的负 Sobolev 正则性下证明了局部适定性。
Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$
本文利用频率分解及改进的 Strichartz 估计，研究实轴上三次 NLS 与 Hartree 方程解的高 Sobolev 范数增长，并改进 Hartree 方程在周期情形下的界。
In this paper, we prove the uniform estimates for the resolvent $(Δ- α)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $α\in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2) \leq q < 2$. In contrast with analogous results on Euclidean space $\mathbb{R}^n$, the exponent $q$ here can be arbitrarily close to $2$. This striking improvement is due to two non-Euclidean features of hyperbolic space: the Kunze-Stein phenomenon and the exponential decay of the spectral measure. In addition, we apply this result to the study of eigenvalue bounds of the Schr…
Growth of Sobolev norms for unbounded perturbations of the Schrödinger equation on flat tori
该研究针对平坦环面上拉普拉斯算子的无界时变扰动问题，证明其Sobolev范数增长满足$\u27e8 t\u27e9^\u03b5$界，为相关偏微分方程解的正则性分析提供了理论结果。
We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is $-Δu=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev embedding theorem, and in some cases the Moser-Trudinger inequality and the Hardy-Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible.
In this paper we consider time dependent Schrödinger equations on the one-dimensional torus $\T := \R /(2 π\Z)$ of the form $\partial_t u = \ii {\cal V}(t)[u]$ where ${\cal V}(t)$ is a time dependent, self-adjoint pseudo-differential operator of the form ${\cal V}(t) = V(t, x) |D|^M + {\cal W}(t)$, $M > 1$, $|D| := \sqrt{- \partial_{xx}}$, $V$ is a smooth function uniformly bounded from below and ${\cal W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schrödinger equation $\partial_t u = \ii {\cal V}(t)[u]$ grow at most…
We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schrödinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schrödinger equations on…
We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators (resp. modified wave operators) on Sobolev spaces from the existence and completeness of the usual wave operators (resp. modified wave operators). We then give some examples of Schrödinger operators to which our abstract result applies. An application to scattering theory for the nonlinear Schrödinger equation with a potential is also given.
Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class
本文研究了在等价对称下自双Chern-Simons-Schrödinger方程的孤子分解问题，证明了在加权Sobolev空间H^{1,1}中，任何解最多分解为一个调制孤子和辐射部分，首次在非完全可积的非线性Schrödinger方程中取得此类结果。
Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces
本文研究了实数轴和圆上三次分数阶非线性 Schrödinger 方程的 Cauchy 问题，通过规范变换与 Duhamel 公式部分迭代处理非线性项，证明了其在负 Sobolev 空间中的整体适定性，并将圆上结果扩展至端点正则性。
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
In this paper we consider linear, time dependent Schrödinger equations of the form ${\rm i} \partial_t ψ= K_0 ψ+ V(t) ψ$, where $K_0$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and $V(t)$ a time periodic potential. We give sufficient conditions on $V(t)$ ensuring that $K_0+V (t)$ generates unbounded orbits. The main condition is that the resonant average of $V(t)$, namely the average with respect to the flow of $K_0$, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations.…
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