A sharp Strichartz estimate for the wave equation with data in the energy space

Neal Bez; Keith M. Rogers

Displaying similar documents to “A sharp Strichartz estimate for the wave equation with data in the energy space”

On bilinear restriction type estimates and applications to nonlinear wave equations

Sergiù Klainerman (1998)

Journées équations aux dérivées partielles

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I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the $L^{2}$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^{p}$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also...

Wave equation and multiplier estimates on ax + b groups

Detlef Müller, Christoph Thiele (2007)

Studia Mathematica

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Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $e^{i t \sqrt L} ψ (\sqrt L / λ)$ for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive...

On the $L_{p}$ -decay and local energy decay of solutions to nonlinear Klein-Gordon equations

Philip Brenner (2003)

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The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes

Jonathan Luk (2013)

Journal of the European Mathematical Society

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We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region ${r \leq \frac{t}{4}}$ .

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

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We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of $ℝ^{3}$ with damping terms of the form ${(- Δ_{x})}^{θ} \partial_{t} u$ , where $θ = 0$ or $θ = 1 / 2$ . The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $θ = 1 / 2$ . For $θ = 0$ existence of smooth attractors is more complicated...

Dispersive and Strichartz estimates on H-type groups

Martin Del Hierro (2005)

Studia Mathematica

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Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as $t^{- p / 2}$ ) and the Schrödinger equation (decay as $t^{(1 - p) / 2}$ ), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

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...

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 ( $η = 0, ψ = 0$ )).

On bilinear estimates for wave equations

Sergiù Klainerman, Damiano Foschi (1999)

Journées équations aux dérivées partielles

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I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the $L^{2}$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^{p}$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also...

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Aissa Guesmia, Salim A. Messaoudi (2005)

Annales Polonici Mathematici

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We consider an initial boundary value problem for the equation $u_{t t} - Δ u - \nabla ϕ \cdot \nabla u + f (u) + g (u_{t}) = 0$ . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2011)

Journal of the European Mathematical Society

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Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let $W$ be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially...

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...

Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière (2014)

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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...

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...

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.

The wave map problem. Small data critical regularity

Igor Rodnianski (2005-2006)

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The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is $(2 + 1)$ , the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory....

The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

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In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1 + 3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p = c^{2} ρ, 0 < c^{2} < 1 / 3$ , the background metric + fluid...