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

Changxing Miao; Youbin Zhu

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Displaying similar documents to “Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure”

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). ...

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s > 1$ . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$ -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c > 0$ such that for any $𝒦 ≫ 1$ we find a solution $u$ and a time $T$ such that ${∥ u (T) ∥}_{H^{s}} \geq 𝒦 {∥ u (0) ∥}_{H^{s}}$ . Moreover, the time $T$ satisfies the polynomial bound $0 < T < 𝒦^{C}$ .

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a graph of order $n$ and $λ (G)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$ . One of the main results of the paper is the following theorem: Let $k \geq 2,$ $n \geq k^{3} + k + 4,$ and let $G$ be a graph of order $n$ , with minimum degree $δ (G) \geq k .$ If $λ (G) \geq n - k - 1,$ then $G$ has a Hamiltonian cycle, unless $G = K_{1} \lor (K_{n - k - 1} + K_{k})$ or $G = K_{k} \lor (K_{n - 2 k} + {\bar{K}}_{k}) .$

Concerning the energy class $_{p}$ for 0 < p < 1

Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)

Annales Polonici Mathematici

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The energy class $_{p}$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of $ℱ_{p}$ and its pluricomplex p-energy is proved.

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

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We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $Δ u - u + f (u) = 0$ in $ℝ^{N}$ , $u \in H^{1} (ℝ^{N})$ , where $N \geq 2$ . Under natural conditions on the nonlinearity $f$ , we prove the existence of $𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠$ in any dimension $N \geq 2$ . Our result complements earlier works of Bartsch and Willem $(N = 4 𝚘𝚛 N \geq 6)$ and Lorca-Ubilla $(N = 5)$ where solutions invariant under the action of $O (2) \times O (N - 2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_{k} \times O (N - 2)$ where $D_{k} \subset O (2)$ denotes the dihedral...

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u (η_{1}, t), \dots, u (η_{q}, t)$ with $0 \leq η_{1} < η_{2} < \dots < η_{q} < 1 .$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $(P_{q})$ of (P) in which the nonlinear term contains the sum $S_{q} [u^{2}] (t) = q^{- 1} \sum_{i = 1}^{q} u^{2} (\frac{(i - 1)}{q}, t)$ . Under suitable conditions, we prove that the solution of $(P_{q})$ converges to the solution...

An upper bound of a generalized upper Hamiltonian number of a graph

Martin Dzúrik (2021)

Archivum Mathematicum

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In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$ -Hamiltonian number of a graph $G$ . We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs $G$ and $H$ using $H$ -Hamiltonian number of $G$ . Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper...

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_{0} \in X$ , where $X \in {M_{2, q}^{s} (ℝ), H^{σ} (𝕋), H^{s_{1}} (ℝ) + H^{s_{2}} (𝕋)}$ and $q \in [1, 2]$ , $s \geq 0$ , or $σ \geq 0$ , or $s_{2} \geq s_{1} \geq 0$ . Moreover, if $M_{2, q}^{s} (ℝ) ↪ L^{3} (ℝ)$ , or if $σ \geq \frac{1}{6}$ , or if $s_{1} \geq \frac{1}{6}$ and $s_{2} > \frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$ . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

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.

Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

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This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation...

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...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E})\dot{z}=F(t,z,u,\dot{u})\,(\in\mathbb{R}^{m})$ with a scalar control $u=u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$ $z(u,\cdot)$ satisfies certain conditions by which $1^{st}$ -order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

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We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_{0} \in H^{s - 1 + ε}$ , $w_{0} \in H^{s - 1}$ and $b_{0} \in H^{s}$ for $s > \frac{3}{2}$ and any $0 < ε < 1$ . The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$ .

On the persistence of decorrelation in the theory of wave turbulence

Anne-Sophie de Suzzoni (2013)

Journées Équations aux dérivées partielles

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We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_{0}$ with values in the Sobolev space $H^{s}$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i θ}$ for all $θ \in ℝ$ . We investigate about the persistence of the decorrelation...