About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard

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Displaying similar documents to “About global existence and asymptotic behavior for two dimensional gravity water waves”

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...

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...

Lyapunov functions and $L^{p}$ -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

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We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $ε c ₜ = k_{c} Δ c - f (c) c + g (a, c)$ , x ∈ Ω, t > 0, for $Ω \subset ℝ^{N}$ , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^{p}$ -estimates.

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...

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

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). ...

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.

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 ${0 \leq t < 1}$

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

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We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f (x, y) = p x + (1 - p) y - \sum_{s = m}^{\infty} s (x, y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s} (x, y) = \sum_{i = 0}^{s} a_{i, s} x^{s - i} y^{i}$ ; (c) $ₘ (1, 1) = \sum_{i = 0}^{m} a_{i, m} > 0$ . Let x₀,x₁ ∈ (0,α) and $x_{n + 1} = f (x ₙ, x_{n - 1})$ , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

Asymptotic integration of differential equations with singular $p$ -Laplacian

Milan Medveď, Eva Pekárková (2016)

Archivum Mathematicum

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In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p -$ Laplacian, where $1 < p < 2$ . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \to \infty$ .

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...

Free decay of solutions to wave equations on a curved background

Serge Alinhac (2005)

Bulletin de la Société Mathématique de France

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We investigate for which metric $g$ (close to the standard metric $g_{0}$ ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (, non integrable) decay conditions on $g - g_{0}$ ; in particular, $g - g_{0}$ decays like $t^{- \frac{1}{2} - ε}$ along wave cones.

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of a positive solution of the system ⎧ $- ∆_{p (x)} u = F (x, u, v)$ , x ∈ Ω, ⎨ $- ∆_{q (x)} v = H (x, u, v)$ , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded domain with C² boundary, $F (x, u, v) = λ^{p (x)} [g (x) a (u) + f (v)]$ , $H (x, u, v) = λ^{q (x}) [g (x) b (v) + h (u)]$ , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $- ∆_{p (x)} u = - {d i v (| \nabla u |}^{p (x) - 2} \nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

The $n$ -th prime asymptotically

Juan Arias de Reyna, Jérémy Toulisse (2013)

Journal de Théorie des Nombres de Bordeaux

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A new derivation of the classic asymptotic expansion of the $n$ -th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{- 1} (n)$ , after having retained the first $m$ terms, for $1 \leq m \leq 11$ , are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_{3}$ such that, for $n \geq r_{3}$ , we have $p_{n} > s_{3} (n)$ where $s_{3} (n)$ is the sum of the first four terms of the asymptotic...

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

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We consider the semilinear Lane–Emden problem where $p > 1$ and $Ω$ is a smooth bounded domain of $ℝ^{2}$ . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of $(_{p})$ , as $p \to + \infty$ . Among other results we show, under some symmetry assumptions on $Ω$ , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p \to + \infty$ , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

On a sum involving the Möbius function

I. Kiuchi, M. Minamide, Y. Tanigawa (2015)

Acta Arithmetica

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Let $c_{q} (n)$ be the Ramanujan sum, i.e. $c_{q} (n) = \sum_{d | (q, n)} d μ (q / d)$ , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for $\sum_{n \leq y} {(\sum_{q \leq x} c_{q} (n))}^{k}$ (k = 1,2) are obtained. As an analogous problem, we evaluate $\sum_{n \leq y} {(\sum_{n \leq x} c ̂_{q} (n))}^{k}$ (k = 1,2), where $c ̂_{q} (n) : = \sum_{d | (q, n)} d | μ (q / d) |$ .