Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

Changxing Miao; Guixiang Xu; Lifeng Zhao

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Displaying similar documents to “Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case”

The defocusing energy-critical Klein-Gordon-Hartree equation

Qianyun Miao, Jiqiang Zheng (2015)

Colloquium Mathematicae

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We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{t t} - Δ u + u + {(| x |}^{- 4} * | u | ²) u = 0$ in spatial dimension d ≥ 5. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the...

On the real analyticity of the scattering operator for the Hartree equation

Changxing Miao, Haigen Wu, Junyong Zhang (2009)

Annales Polonici Mathematici

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We study the real analyticity of the scattering operator for the Hartree equation $i \partial_{t} u = - Δ u + u (V * | u | ²)$ . To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the...

Dynamics of a modified Davey-Stewartson system in ℝ³

Jing Lu (2016)

Colloquium Mathematicae

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We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i \partial ₜ u + Δ u = λ ₁ | u | ⁴ u + λ ₂ b ₁ u v_{x ₁}$ , $- Δ v = b ₂ {(| u | ²)}_{x ₁}$ . Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].

On blow-up for the Hartree equation

Jiqiang Zheng (2012)

Colloquium Mathematicae

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We study the blow-up of solutions to the focusing Hartree equation $i u_{t} + Δ u + {(| x |}^{- γ} * | u | ²) u = 0$ . We use the strategy derived from the almost finite speed of propagation ideas devised by Bourgain (1999) and virial analysis to deduce that the solution with negative energy (E(u₀) < 0) blows up in either finite or infinite time. We also show a result similar to one of Holmer and Roudenko (2010) for the Schrödinger equations using techniques from scattering theory.

On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev (1992)

Annales de l'institut Fourier

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We consider selfadjoint positively definite operators of the form $- Δ + P$ (not necessarily elliptic) in $ℝ^{n}$ , $n \geq 3$ , odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if ${λ_{j}} (Im λ_{j} \geq 0_{)}$ are the scattering poles associated to the operator $- Δ + P$ repeated according to multiplicity, it is proved that for any $ϵ > 0$ there exists...

Recovering Asymptotics at Infinity of Perturbations of Stratified Media

Tanya Christiansen, Mark S. Joshi (2000)

Journées équations aux dérivées partielles

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We consider perturbations of a stratified medium $ℝ_{x}^{n - 1} \times ℝ_{y}$ , where the operator studied is $c^{2} (x, y) Δ$ . The function $c$ is a perturbation of $c_{0} (y)$ , which is constant for sufficiently large $| y |$ and satisfies some other conditions. Under certain restrictions on the perturbation $c$ , we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of $c$ from knowledge of $c_{0}$ and the singularities of the scattering matrix at...

Propagation of singularities in many-body scattering in the presence of bound states

András Vasy (1999)

Journées équations aux dérivées partielles

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In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u \in 𝒮^{'}$ of $(H - λ) u = 0$ , where $H$ is a many-body hamiltonian $H = Δ + V$ , $Δ \geq 0$ , $V = \sum_{a} V_{a}$ , and $λ$ is not a threshold of $H$ , under the assumption that the inter-particle (e.g. two-body) interactions $V_{a}$ are real-valued polyhomogeneous symbols of order $- 1$ (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is...

Absolutely continuous and singular spectral shift functions

Nurulla Azamov

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Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{\pm} (λ; H ₀ + r V, H ₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including...

Stability of the inverse problem in potential scattering at fixed energy

Plamen Stefanov (1990)

Annales de l'institut Fourier

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We prove an estimate of the kind $∥ q_{1} - q_{2} ∥_{L^{\infty}} \leq C ϕ (∥ A_{q_{1}} - A_{q_{2}} ∥_{R, 3 / 2 - 1 / 2})$ , where $A_{q_{i}} (ω, θ)$ , $i = 1, 2$ is the scattering amplitude related to the compactly supported potential $q_{i} (x)$ at a fixed energy level $k =$ const., $ϕ (t) = (- ln t)^{- δ}$ , $0 < δ < 1$ and $∥ \cdot ∥_{R, 3 / 2 - 1 / 2}$ is a suitably defined norm.

Scattering for 1D cubic NLS and singular vortex dynamics

Valeria Banica, Luis Vega (2012)

Journal of the European Mathematical Society

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We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions $χ_{a} (t, x)$ form a family of evolving regular curves in $ℝ^{3}$ that develop a singularity in finite time, indexed by a parameter $a > 0$ . We consider curves that are small regular perturbations of $χ_{a} (t_{0}, x)$ for a fixed time $t_{0}$ . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of...

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2011)

Journal of the European Mathematical Society

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Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let $W$ be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially...

Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2012)

Journal of the European Mathematical Society

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Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.

The $n$ -centre problem of celestial mechanics for large energies

Andreas Knauf (2002)

Journal of the European Mathematical Society

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We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all energies $E$ above a positive threshold. Whereas for $n = 1$ there are no bounded orbits, and for $n = 2$ there is just one closed orbit, for $n \geq 3$ the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of...

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong (2017)

Czechoslovak Mathematical Journal

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The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p, 1}^{n / p - 1} (ℝ^{n}) \times {\dot{B}}_{p, 1}^{n / p} (ℝ^{n})$ with $n < p < 2 n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

Annales de l'institut Fourier

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Let $S (λ)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪 \subset R^{n}$ , $n \geq 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$ . The function $s (λ)$ , called scattering phase, is determined from the equality $e^{- 2 π i s (λ)} = \det S (λ)$ . We show that $s (λ)$ has an asymptotic expansion $s (λ) \sim \sum_{j = 0}^{\infty} c_{j} λ^{n - j}$ as $λ \to + \infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

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We consider the mass critical (gKdV) equation $u_{t} + {(u_{x x} + u^{5})}_{x} = 0$ for initial data in $H^{1}$ . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].