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

Jonathan Luk

Displaying similar documents to “The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes”

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...

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

Neal Bez, Keith M. Rogers (2013)

Journal of the European Mathematical Society

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We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L_{t, x}^{4} (ℝ^{5 + 1})$ norm of the solution in terms of the energy. We also characterise the maximisers.

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...

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...

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

Philip Brenner (2003)

Banach Center Publications

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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...

Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

Frank Merle, Hatem Zaag (2009-2010)

Séminaire Équations aux dérivées partielles

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We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u (x, t)$ , the graph $x \mapsto T (x)$ of its blow-up points and $𝒮 \subset ℝ$ the set of all characteristic points and show that $𝒮$ is locally finite. Finally, given $x_{0} \in 𝒮$ , we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that...

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.

Hyperbolic Equations in Uniform Spaces

J. W. Cholewa, Tomasz Dlotko (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ℝ ⁿ. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space $H ̇ ¹_{l u} (ℝ ⁿ) \times L ̇ ²_{l u} (ℝ ⁿ)$ . Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.

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.

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

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2012)

Journal of the European Mathematical Society

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Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.

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...

The wave map problem. Small data critical regularity

Igor Rodnianski (2005-2006)

Séminaire Bourbaki

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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....

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...

Radiation fields

Piotr T. Chruściel, Olivier Lengard (2005)

Bulletin de la Société Mathématique de France

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We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $λ φ^{p}$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times...

The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

Ferruccio Colombini, Guy Métivier (2008)

Annales scientifiques de l'École Normale Supérieure

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In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form $L u : = \sum_{j, k = 0}^{n} \partial_{y_{j}} (a_{j, k} \partial_{y_{k}} u) + \sum_{j = 0}^{n} {b_{j} \partial_{y_{j}} u + \partial_{y_{j}} (c_{j} u)} + d u = f,$ when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides...