Weighted fractional chain rule and nonlinear wave equations with minimal regularity

Abstract

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data:It has been known that the problem is well-posed for s ≥ 2 and ill-posed for s < 3/2. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for s > 3/2 and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.1.1. Quadratic semi-linear wave equation . In [19], by using bilinear estimates, together with standard energy-type estimates, Klainerman and Machedon proved the local well-posedness for the problem (1.1) with a + b = 0 (thus, satisfying the null condition) in H 2 , which was later improved to H s for s > s c , see [20], [21], [22] and references therein. Moreover, even without the null condition, their bilinear

Blow up for small-amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds

“…it turns out that the homogeneous local energy and KSS-type estimates are available (see Metcalfe-Sogge [22] and Hidano-Wang-Yokoyama [10, Lemma 2.3]). Thus, in spirit of [9] and [7], we could prove well-posed results for the problem with c 2 = 0 in the radial case, with sharp lower bound of the lifespan.…”

“…These nonlinear estimates are well-known, see, e.g., [9] and [7]. For reader's convenience, we give a proof of (2.12) here.…”

“…The Glassey conjecture for small, radial asymptotically Euclidean manifolds. As we mentioned before, when g = g 0 and n ≥ 2, the local wellposedness and long time existence for (1.15) with b = c 2 = 0 has been studied in [9] and [7], by exploiting Morawetz type local energy estimates and its variants (see, e.g., [15], [22], [10]), together with the trace estimates. The similar approach could apply in the setting of small, radial asymptotically Euclidean manifolds, at least when n ≥ 3.…”

“…The high dimensional existence part remains open, except for the radial case, see Hidano-Wang-Yokoyama [9]. The upper bound (1.10) is also known to be sharp in general, at least for radial data, see Hidano-Jiang-Wang-Lee [7] and references therein.…”

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

“…It should be mentioned that improvement of regularity in the presence of radial symmetry was first observed in [18] for semilinear equations such as ✷u = c 1 (∂ t u) 2 + c 2 |∇u| 2 . (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].…”