Generalized and weighted Strichartz estimates

Abstract

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied Analysis. 33 pages. 2 more references adde

Space-time L 2 estimates, regularity and almost global existence for elastic waves

“…(See also page 176 of [31], Section 5 of [25], and [9], [30], and [6] for related results.) We also mention that it was also observed for the wave equation with a power-type nonlinear term ✷u = F (u), first by Lindbald and Sogge [23], and then by some authors (see [4], [12], [3]), and quasilinear wave equations of the form ∂ 2 t u − a 2 (u)∆u = c 1 (∂ t u) 2 + c 2 |∇u| 2 , by [7] and [41]. In particular, the almost global existence of low regularity radially symmetric solutions with small initial data was showed in [9] and [7] for the semilinear and the quasilinear case in 3-D, respectively.…”

Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

“…It is not difficult to find that the lower bound of the lifespan of the critical problem of (1.2) is closely related to the power q of L q t in time norm for the forcing term in the key estimates. For example, the obtained bound exp cε −qc(qc−1) which comes from [14] coincides with q = q c in estimates (3.1), and the obtained bound exp cε −(qc−1) 2 /2 which comes from [11] coincides with q = (q c − 1)/2 in estimates (3.2). In order to improve the result from [9], we adapt these generalized Strichartz estimates to the equation (1.1), use energy inequality with Klainerman-Sobolev inequality to deal with derivative term.…”

“…Then there exists a constant C, such that ∀ T < T * (3.1) [11]). When n = 2, suppose u solves the linear equation…”

“…Instead, the approach using space-time estimates is more robust for various nonlinear problems. Here, we adapt the generalized Strichartz estimates from [11] to the current setting and obtain the following…”

On the boundary Strichartz estimates for wave and Schrödinger equations

“…It was known that (1.8) allows a wider range of indices (q, p) than (1.2). For the wave equation a = 1 and d ≥ 2, the optimal range of (q, p) for (1.8) is (see [9] and references therein, [17] for d = 2): (q, p) = (∞, 2) or…”