Free decay of solutions to wave equations on a curved background

Serge Alinhac

Displaying similar documents to “Free decay of solutions to wave equations on a curved background”

Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc Joly, Guy Métivier, Jeffrey Rauch (1998-1999)

Séminaire Équations aux dérivées partielles

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The nonlinear dissipative wave equation $u_{t t} - Δ u + {| u_{t} |}^{h - 1} u_{t} = 0$ in dimension $d > 1$ has strong solutions with the following structure. In $0 \leq t < 1$ the solutions have a focusing wave of singularity on the incoming light cone $| x | = 1 - t$ . In ${t \geq 1}$ that is after the focusing time, they are smoother than they were in ${0 \leq t < 1}$ . The examples are radial and piecewise smooth in ${0 \leq t < 1}$

Modeling of the resonance of an acoustic wave in a torus

Jérôme Adou, Adama Coulibaly, Narcisse Dakouri (2013)

Annales mathématiques Blaise Pascal

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A pneumatic tyre in rotating motion with a constant angular velocity $Ω$ is assimilated to a torus whose generating circle has a radius $R$ . The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$ . When $(Ω R / C_{0}) ≪ 1$ and $(a / R) ≪ 1$ (where $C_{0}$ is the velocity of the sound), we show that at the rapid time scale $R / C_{0}$ , the air motion within a torus periodically excited on its surface generates an acoustic wave $h$ . A study of this acoustic wave is conducted and shows that the mode associated...

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

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We consider the damped wave equation $α u_{t t} + u_{t} = u_{x x} - V^{'} (u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u (x, t) = h (x - s t)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$ . We show that, if the initial data are sufficiently close to the profile of a front for large $| x |$ , the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t \to + \infty$ . The proof of this...

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

Studia Mathematica

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Large time behavior of solutions to the generalized damped wave equation $u_{t t} + A u_{t} + ν B u + F (x, t, u, u_{t}, \nabla u) = 0$ for $(x, t) \in ℝ^{n} \times [0, \infty)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A u_{t} = u_{t}$ , $B u = - Δ u$ , and the nonlinear term is either $| u_{t} |^{q - 1} u_{t}$ or ${| u |}^{α - 1} u$ . In this case, the asymptotic profile of solutions...

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...

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

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Let WF⁎ be the wave front set with respect to $C^{\infty}$ , quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C ₀^{\infty}$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u *_{B} φ (x) = \int u (x - y) φ (y) B (x, y) d y$ , where $B \in C^{\infty}$ is appropriate, and prove that if $(u *_{B} φ, φ) \geq 0$ for every $φ \in C ₀^{\infty}$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

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...

Breathers for nonlinear wave equations

Michael W. Smiley (1988)

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

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The semilinear differential equation (1), (2), (3), in $\mathbb{R}\times\Omega$ with $\Omega\in\mathbb{R}^{N}$ , (a nonlinear wave equation) is studied. In particular for $\Omega=\mathbb{R}^{3}$ , the existence is shown of a weak solution $u(t,x)$ , periodic with period $T$ , non-constant with respect to $t$ , and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu(t,|x|)$ . The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative...

Some results on spaces with $ℵ_{1}$ -calibre

Wei-Feng Xuan, Wei-Xue Shi (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove that, assuming , if $X$ is a space with $ℵ_{1}$ -calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{δ}$ -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with $ℵ_{1}$ -calibre.

Stability in exponential time of Minkowski space-time with a space-like translation symmetry

Cécile Huneau (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the $3 + 1$ vacuum Einstein equations reduce to the $2 + 1$ Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in $1 / \sqrt{t}$ of free solutions...

Breathers for nonlinear wave equations

Michael W. Smiley (1988)

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

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On multiset colorings of generalized corona graphs

Yun Feng, Wensong Lin (2016)

Mathematica Bohemica

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A vertex $k$ -coloring of a graph $G$ is a if $M (u) \neq M (v)$ for every edge $u v \in E (G)$ , where $M (u)$ and $M (v)$ denote the multisets of colors of the neighbors of $u$ and $v$ , respectively. The minimum $k$ for which $G$ has a multiset $k$ -coloring is the $χ_{m} (G)$ of $G$ . For an integer $ℓ \geq 0$ , the $ℓ$ - of a graph $G$ , ${cor}^{ℓ} (G)$ , is the graph obtained from $G$ by adding, for each vertex $v$ in $G$ , $ℓ$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for $ℓ$ - of all complete graphs, the regular complete...

Oscillating multipliers on the Heisenberg group

E. K. Narayanan, S. Thangavelu (2001)

Colloquium Mathematicae

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Let ℒ be the sublaplacian on the Heisenberg group Hⁿ. A recent result of Müller and Stein shows that the operator $ℒ^{- 1 / 2} s i n \sqrt ℒ$ is bounded on $L^{p} (H ⁿ)$ for all p satisfying |1/p - 1/2| < 1/(2n). In this paper we show that the same operator is bounded on $L^{p}$ in the bigger range |1/p - 1/2| < 1/(2n-1) if we consider only functions which are band limited in the central variable.

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let $ϕ$ be a substitution of Pisot type on the alphabet $𝒜 = {1, 2, ..., d}$ ; $ϕ$ satisfies theif for every $i, j \in 𝒜$ , there are integers $k, n$ such that $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same $k$ -th letter, and the prefixes of length $k - 1$ of $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d = 2$ and provide a partial result for $d \geq 2$ .

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, G. Maillard (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study the parabolic Anderson equation $\partial u (x, t) / \partial t = κ 𝛥 u (x, t) + ξ (x, t) u (x, t)$ , $x \in ℤ^{d}$ , $t \geq 0$ , where the $u$ -field and the $ξ$ -field are $ℝ$ -valued, $κ \in [0, \infty)$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $ξ$ -field plays the role of athat drives the equation. The initial condition $u (x, 0) = u_{0} (x)$ , $x \in ℤ^{d}$ , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2 d κ$ ,...

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). ...

Cutting the loss of derivatives for solvability under condition $(Ψ)$

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition $(ψ)$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $ϵ + 3 / 2$ for any $ϵ > 0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(ψ)$ doesimply local solvability with a loss of 1...