Axel Gruenrock
We prove some local (in time) wellposedness results for nonlinear Schroedinger equations with rough data, that is, the initial value belongs to some Sobolev space of negative index. The proof uses the Fourier restriction norm method.
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 two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere. The second one provides local wellposedness for quadratic nonlinearities in the case of zonal data on the sphere. Both results are based on new multilinear Strichartz-type estimates for the Schroedinger group.
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…
Global wellposedness for the energy-critical Zakharov system below the ground state
本文研究四维能量临界Zakharov系统的柯西问题，借助带势薛定谔方程的一致Strichartz估计及新的双线性傅里叶限制估计，证明能量和波质量低于基态阈值的非径向初值在全能量空间中全局适定。
We consider a Cauchy problem of energy-critical fractional Schrödinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition, introduced by Á. Bényi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schrödinger equation ($α= 2$).
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.
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain Ω. First, we prove an appropriate Strichartz type estimate using the $L^q$ spectral projector estimates of the Laplace operator. Our proof follows Burq-Lebeau-Planchon [5]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a qu…
We present some a - priori bounds from above and from below for solutions to a class of conformally invariant Schroedinger equations. As a by - product we deduce some new uniqueness results.
The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$
本文通过新的$L^4$-Strichartz估计和双线性Strichartz估计，解决了周期性KP-II方程在负Sobolev正则性初始数据下的Cauchy问题。
We consider the Cauchy problem for the nonlinear Schrödinger equation with double nonlinearities with opposite sign, with one term is mass-critical and the other term is mass-supercritical and energy-subcritical, which includes the famous two-dimensional cubic-quintic nonlinear Schrödinger equaton. We prove global wellposedness and scattering in $H^1(\mathbb{R}^d)$ below the threshold for non-radial data when $1 \le d \le 4$.
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schrödinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} < \|Q\|_{L^2} \|\nabla Q\|_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation…
We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. S…
We prove smoothing estimates for Schrödinger equations $i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.
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。
We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the ``moving charge'' of the spinor. This limit, sometimes also called ``Post-Newtonian'' limit, yields a Schroedinger-Poisson system, where the spin and the magnetic field no longer appear. However, our splitting of the 4-spinor into two 2-spinors preserves the symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger equati…
We study the quantum mechanical many-body problem of $N$ nonrelativistic electrons interacting with their self-generated classical electromagnetic field and $K$ static nuclei. The system of coupled equations governing the dynamics of the electrons and their self-generated electromagnetic field is referred to as the many-body Maxwell-Pauli equations. Here we construct time global, finite-energy, weak solutions to the many-body Maxwell-Pauli equations under the assumption that the fine structure constant $α$ and the atomic numbers are not too large. The particular assumptions on the size of $α$ …
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。