Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Jean Bourgain; Aynur Bulut

Displaying similar documents to “Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case”

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

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Nonlinear Schrödinger equations (NLS) $_{a}$ with strongly singular potential ${a | x |}^{- 2}$ on a bounded domain $Ω$ are considered. If $Ω = ℝ^{N}$ and $a > - {(N - 2)}^{2} / 4$ , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a = - {(N - 2)}^{2} / 4$ is excluded because $D (P_{a (N)}^{1 / 2})$ is not equal to $H^{1} (ℝ^{N})$ , where $P_{a (N)} : = - Δ - {(N - 2)}^{2} / {(4 | x |}^{2})$ is nonnegative and selfadjoint in $L^{2} (ℝ^{N})$ . On the other hand, if $Ω$ is a smooth and bounded domain with $0 \in Ω$ , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

Invariance of the Gibbs measure for the Benjamin–Ono equation

Yu Deng (2015)

Journal of the European Mathematical Society

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In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^{2}$ , extending the $L^{2}$ well-posedness result of Molinet [20].

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani (2012)

Journal of the European Mathematical Society

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We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space $ℱ L^{s, r} (T)$ with $s \geq \frac{1}{2}, 2 < r < 4, (s - 1) r < - 1$ and scaling like $H^{\frac{1}{2} - ϵ} (𝕋)$ , for small $ϵ > 0$ . We also show the invariance of this measure.

Global well-posedness for the Klein-Gordon-Schrödinger system with higher order coupling

Agus Leonardi Soenjaya (2022)

Mathematica Bohemica

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Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u, n) \in L^{2} \times L^{2}$ under some conditions on the nonlinearity (the coupling term), by using the $L^{2}$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman, Fabricio Macià (2014)

Journal of the European Mathematical Society

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In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality,...

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V (x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $ε$ and the number of positive solutions increases to $\infty$ as $ε \to 0$ . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V (x)$ . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov (2013)

Annales de l’institut Fourier

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In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^{2}$ critical...

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

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In the half-space $ℝ^{d} \times ℝ ₊$ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$ . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

On the radius of spatial analyticity for the higher order nonlinear dispersive equation

Aissa Boukarou, Kaddour Guerbati, Khaled Zennir (2022)

Mathematica Bohemica

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In this work, using bilinear estimates in Bourgain type spaces, we prove the local existence of a solution to a higher order nonlinear dispersive equation on the line for analytic initial data $u_{0}$ . The analytic initial data can be extended as holomorphic functions in a strip around the $x$ -axis. By Gevrey approximate conservation law, we prove the existence of the global solutions, which improve earlier results of Z. Zhang, Z. Liu, M. Sun, S. Li, (2019).

Waves in Honeycomb Structures

Charles L. Fefferman, Michael I. Weinstein (2012)

Journées Équations aux dérivées partielles

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We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, $V$ . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_{V} = - Δ + V$ and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution $e^{- i H_{V} t} ψ_{0}$ , for data $ψ_{0}$ , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion...

Global existence of solutions to Schrödinger equations on compact riemannian manifolds below $H^{1}$

Sijia Zhong (2010)

Bulletin de la Société Mathématique de France

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In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. $s < 1$ , under some bilinear Strichartz assumption. We will find some $\tilde{s} < 1$ , such that the solution is global for $s > \tilde{s}$ .

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

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We investigate the propagation of the uniform spatial Gevrey $G^{σ}$ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

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We construct the fundamental solution of $\partial_{t} - Δ_{y} - q (t, y)$ for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure...

A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio, David Ruiz (2015)

Journal of the European Mathematical Society

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This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $- Δ u (x) + (ω + \frac{h^{2} (| x |)}{{| x |}^{2}} + \int_{| x |}^{+ \infty} \frac{h (s)}{s} u^{2} (s) d s) u (x) = {| u (x) |}^{p - 1} u (x)$ , where $h (r) = \frac{1}{2} \int_{0}^{r} s u^{2} (s) d s$ . This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p \in (1, 3)$ , the functional may be bounded from below or not, depending on $ω$ . Quite surprisingly, the threshold value for $ω$ is explicit....

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

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We also prove a long time existence result; more precisely we prove that for fixed $T > 0$ there exists a set $Σ_{T}$ , $ℙ (Σ_{T}) > 0$ such that any data $φ^{ω} (x) \in H^{γ} (𝕋^{3}), γ < 1, ω \in Σ_{T}$ , evolves up to time $T$ into a solution $u (t)$ with $u (t) - e^{i t Δ} φ^{ω} \in C ([0, T]; H^{s} (𝕋^{3}))$ , $s = s (γ) > 1$ . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^{1} (𝕋^{3})$ , that is in the supercritical scaling regime.

On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra, Lotfi Riahi (2004)

Annales Polonici Mathematici

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We consider the general Schrödinger operator $L = d i v (A (x) \nabla_{x}) - μ$ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function $G_{Δ}$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K ₙ^{\infty}$ considered by Zhao and Pinchover. As an application we study the cone $_{L} (ℝ ⁿ ₊)$ of all positive L-solutions continuously...

The cubic nonlinear Dirac equation

Federico Cacciafesta (2012)

Journées Équations aux dérivées partielles

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We present some results obtained in collaboration with prof. Piero D’Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in $H^{1}$ with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V (x) \sim {| x |}^{- α}$ , $0 < α < 2$ , and $K (x) \sim {| x |}^{- β}$ , $β > 0$ . Working in weighted Sobolev spaces, the existence of ground states $v_{ε}$ belonging to $W^{1, 2} (ℝ^{N})$ is proved under the assumption that $σ < p < (N + 2) / (N - 2)$ for some $σ = σ_{N, α, β}$ . Furthermore, it is shown that $v_{ε}$ are spikes concentrating at a minimum point of $𝒜 = V^{θ} K^{- 2 / (p - 1)}$ , where $θ = (p + 1) / (p - 1) - 1 / 2$ .

On the $L_{p}$ -decay and local energy decay of solutions to nonlinear Klein-Gordon equations

Philip Brenner (2003)

Banach Center Publications

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Quasi-periodic solutions with Sobolev regularity of NLS on $𝕋^{d}$ with a multiplicative potential

Massimiliano Berti, Philippe Bolle (2013)

Journal of the European Mathematical Society

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We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $𝕋^{d}, d \geq 1$ , finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty}$ then the solutions are $C^{\infty}$ . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized...