Generalized Strichartz Estimates Research Articles

Generalized Strichartz estimates for wave and Dirac equations in Aharonov–Bohm magnetic fields

We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.

Local and global existence in Lp for the inhomogeneous nonlinear Schrödinger equation

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|−b|u|βu. It is show that a global solution exists in the mass‐subcritical for large data in the spaces Lp, p < 2 under some suitable conditions on b, β, and p. The solution is established using a data‐decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

Randomized final-state problem for the Zakharov system in dimension three

We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on and an angular randomization on yielding random final states We obtain that for almost every ω, there is a unique solution of the Zakharov system scattering to the final state The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.

Decay of the local energy for the solutions of the critical Klein–Gordon equation

We prove the exponential decay of local energy for the Klein–Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax–Phillips.

Generalized inhomogeneous Strichartz estimates

We prove new inhomogeneous generalized Strichartz estimates, which do not follow from the homogeneous generalized estimates by virtue of the Christ-Kiselev lemma. Instead, we make use of the bilinear interpolation argument worked out by Keel and Tao and refined by Foschi presented in a unified framework. Finally, we give a sample application.

Sharp spherically averaged Strichartz estimates for the Schrödinger equation

We prove generalized Strichartz estimates with weaker angular integrability for the Schrödinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in Guo Z et al (2014 Generalized Strichartz estimates and scattering for 3D Zakharov system Commun. Math. Phys. 331 239–59).

Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

For any subcritical index of regularity s > 3/2, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space H s × H s−1 with certain angular regularity. The lifespan is known to be sharp in general. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space . In the last section, we also consider the general semilinear wave equations with the spatial dimension n ≥ 2 and the order of nonlinearity p ≥ 3.

The Glassey conjecture with radially symmetric data

In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions. Moreover, we are able to prove the existence results with low-regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates, KSS type estimates, and the generalized Strichartz estimates.

Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).

Scattering for the critical and localised semilinear wave equation

In this paper, we prove a scattering theorem for the critical wave equation outside convex obstacle. The proof relies on generalized Strichartz estimates.