Radiation fields

Piotr T. Chruściel; Olivier Lengard

Displaying similar documents to “Radiation fields”

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

Thomas Alazard (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^{2}$ and $L^{\infty}$ estimates. The $L^{2}$ bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates...

An invariance method for constructing fundamental solutions for $P (□_{m} n)$

Zofia Szmydt, Bogdan Ziemian (1985)

Annales Polonici Mathematici

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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...

Separation properties for self-conformal sets

Yuan-Ling Ye (2002)

Studia Mathematica

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For a one-to-one self-conformal contractive system ${w_{j}}_{j = 1}^{m}$ on $ℝ^{d}$ with attractor K and conformality dimension α, Peres et al. showed that the open set condition and strong open set condition are both equivalent to $0 < ℋ^{α} (K) < \infty$ . We give a simple proof of this result as well as discuss some further properties related to the separation condition.

Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

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We study the Cauchy problem in the hyperbolic space $ℍ^{n} (n \geq 2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space $ℝ^{n}$ new phenomena arise, which depend on the properties of the diffusion process in $ℍ^{n}$ . We also investigate a family of travelling wave solutions,...

Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière (2014)

Annales de l’institut Fourier

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We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of $β$ -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of $β$ -damped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic...

Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation

Rémi Carles, David Lannes (2003)

Bulletin de la Société Mathématique de France

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We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like $ln ε$ , where $ε$ stands for...

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...

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...

On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (2013)

Annales scientifiques de l'École Normale Supérieure

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In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X, d)$ , and its conformal dimension ${dim}_{A R} (X, d)$ . Using a sequence of finite coverings of $(X, d)$ , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute ${dim}_{A R} (X, d)$ using the critical exponent $Q_{N}$ associated to the combinatorial modulus.

Large data local solutions for the derivative NLS equation

Ioan Bejenaru, Daniel Tataru (2008)

Journal of the European Mathematical Society

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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 $n \geq 2$ .

Lower quantization coefficient and the F-conformal measure

Mrinal Kanti Roychowdhury (2011)

Colloquium Mathematicae

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Let $F = f^{(i)} : 1 \leq i \leq N$ be a family of Hölder continuous functions and let $φ_{i} : 1 \leq i \leq N$ be a conformal iterated function system. Lindsay and Mauldin’s paper [Nonlinearity 15 (2002)] left an open question whether the lower quantization coefficient for the F-conformal measure on a conformal iterated funcion system satisfying the open set condition is positive. This question was positively answered by Zhu. The goal of this paper is to present a different proof of this result.

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 ${0 \leq t < 1}$

Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Oana Ivanovici (2012)

Journal of the European Mathematical Society

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In this paper we consider a smooth and bounded domain $Ω \subset ℝ^{d}$ of dimension $d \geq 2$ with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex...

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.

Bounded Solutions for the Degasperis-Procesi Equation

Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)

Bollettino dell'Unione Matematica Italiana

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This paper deals with the well-posedness in $L^{1}\cap L^{\infty}$ of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.