Related papers: Sharp dispersive estimates for the wave equation o…
In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove…
We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$ dispersive decay rate is $|t|^{-3/4}$ for $d=2$, $|t|^{-7/6}$ for $d=3$ and…
Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical…
We establish the sharp \( l^1 \to l^{\infty} \) decay estimate for the discrete Schr\"odinger equation (DS) on the Layered King's Grid (LKG), with a dispersive decay rate of \( \langle t \rangle^{-13/12} \), which is faster than that for…
We prove sharper Strichartz estimates than expected from theoptimal dispersion bounds.
In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$…
The discrete Schr\"odinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate…
In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…
We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix…
We prove global Strichartz estimates without loss for the wave equation outside two strictly convex obstacles, following the roadmap introduced in [Lafontaine, 2017] for the Schr\"odinger equation. Moreover, we show a first step toward the…
We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L^4_{t,x}(\R^{5+1})$ norm of the solution in terms of the energy. We also characterise the…
Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales…
We consider the sharp Strichartz estimate for the wave equation on $\mathbb R^{1+5}$ in the energy space, due to Bez and Rogers. We show that it can be refined by adding a term proportional to the distance from the set of maximisers, in the…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension 1 in presence of a flat bottom. We prove a decay with respect to time t of order 1/3 for solutions with initial data in weighted Sobolev…
We prove the dispersive and Strichartz estimates for solutions to the wave equation with a class of many-electric potentials in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the…
In this paper we show a general Strichartz estimate for certain perturbed wave equation, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay…
We consider the wave equation on a manifold $(\Omega,g)$ of dimension $d\geq 2$ with smooth strictly convex boundary $\partial\Omega\neq\emptyset$, with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then…
For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher…
We calculate the the sharp constant and characterise the extremal initial data in $\dot{H}^{\frac{3}{4}}\times\dot{H}^{-\frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four space dimensions.