Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system

Piero D'Ancona; Damiano Foschi; Sigmund Selberg

Displaying similar documents to “Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system”

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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Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $\partial_{t} u - \partial ³_{t x x} u + 2 κ \partial_{x} u + \partial_{x} [g (u) / 2] = γ (2 \partial_{x} u \partial ²_{x x} u + u \partial ³_{x x x} u)$ for the initial data u₀(x) in the Besov space $B_{p, r}^{s} (ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$ -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

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We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$ . Here we prove a similar result for large initial data in all dimensions $n \geq 2$ .

Strichartz estimates for water waves

Thomas Alazard, Nicolas Burq, Claude Zuily (2011)

Annales scientifiques de l'École Normale Supérieure

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In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ( $η = 0, ψ = 0$ )).

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $P$ be a long range metric perturbation of the Euclidean Laplacian on $ℝ^{d}$ , $d \geq 2$ . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$ . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{i t f (P)}$ where $f$ has a suitable development at zero (resp. infinity). ...

Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao, Youbin Zhu (2006)

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We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ $u_{t t} - Δ u + m ² u = - F ₁ (| u | ², | v | ²) u$ , ⎨ ⎩ $v_{t t} - Δ v + m ² v = - F ₂ (| u | ², | v | ²) v$ for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies $E (u, v, ℝ ⁿ, 0) = 1 / 2 \int_{ℝ ⁿ} (| \nabla u (0) | ² + | u_{t} (0) | ² + m ² | u (0) | ² + | \nabla v (0) | ² + | v_{t} (0) | ² + m ² | v (0) | ² + F (| u (0) | ², | v (0) | ²)) d x < \infty$ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a >...

Weakly regular $T^{2}$ -symmetric spacetimes. The global geometry of future Cauchy developments

Philippe LeFloch, Jacques Smulevici (2015)

Journal of the European Mathematical Society

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We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with $T^{2}$ -symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties...

A regularity criterion for the 2D MHD and viscoelastic fluid equations

Zhuan Ye (2015)

Annales Polonici Mathematici

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This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies $\nabla B, \nabla F \in L^{q} (0, T; L^{p} (ℝ ²))$ for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

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We consider the maximal function $| | (S^{a} f) [x] {| |}_{L^{\infty} [- 1, 1]}$ where $(S^{a} f) {(t)}^{\land} (ξ) = e^{{i t | ξ |}^{a}} f ̂ (ξ)$ and 0 < a < 1. We prove the global estimate $| | S^{a} f {| |}_{L ² (ℝ, L^{\infty} [- 1, 1])} {\leq C | | f | |}_{H^{s} (ℝ)}$ , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

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This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $Ω \subset ℝ^{n}$ , with Dirichlet boundary conditions. The observation is done on a subset $ω$ of Lebesgue measure $| ω | = L | Ω |$ , where $L \in (0, 1)$ is fixed. We denote...

Local connectedness, connectedness im kleinen, and another properties of hyperspaces of $R_{0}$ spaces

C. Dorsett (1979)

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Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2016)

Journal of the European Mathematical Society

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We consider the wave and Schrödinger equations on a bounded open connected subset $Ω$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $ω$ of $Ω$ during a time interval $[0, T]$ with $T > 0$ . It is well known that, if the pair $(ω, T)$ satisfies the Geometric Control Condition ( $ω$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions...

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ℝ^{3}$

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of $H = - Δ + i (A \cdot \nabla + \nabla \cdot A) + V = : - Δ + L 1$ for large energies. It is shown here that there exist a large integer $m$ and a large number $λ_{0}$ so that relative to the usual weighted $L^{2}$ -norm, $∥ {(L {(- Δ + (λ + i 0))}^{- 1})}^{m} ∥ < \frac{1}{2} 2$ for all $λ > λ_{0}$ . This requires suitable decay and smoothness conditions on $A, V$ . The estimate (2) is trivial when $A = 0$ , but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

Teo Mora (1989)

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A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function $f \in W^{k, p} (Ω)$ possesses an $L^{p}$ derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space $W^{k, p} (Ω)$ . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1<p<2$ : an a priori estimate

Andrea Gentile (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form $\mathcal{F}(\nu,\Omega)=\int_{\Omega}f(x,D\nu(x))\,dx$ with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Explicit estimates on the summatory functions of the Möbius function with coprimality restrictions

Olivier Ramaré (2014)

Acta Arithmetica

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We prove that $| \sum_{d \leq x, (d, q) = 1} μ (d) / d | \leq 2.4 (q / φ (q)) / l o g (x / q)$ for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,

On connectivity properties of hyperspaces of $R_{0}$ spaces

M. Mršević (1979)

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Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

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This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012). We address the decay rates of the energy of the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger...

On the global regularity of $N$ -dimensional generalized Boussinesq system

Kazuo Yamazaki (2015)

Applications of Mathematics

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We study the $N$ -dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.