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: $\square u=a|{\partial }_{t}u{|}^2+b|{\nabla }_{x}u{|}^2$, $u(0,x)=u_0(x)\in H_{\mathrm{rad}}^s$, ${\partial }_{t}u(0,x)=u_1(x)\in H_{\mathrm{rad}}^{s-1}$. It has been known that the problem is well-posed for $s\ge 2$ 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.

14 pages. The previous well-posed results have been strengthened to unconditional well-posedness

References in corpus (5)

• The Strauss conjecture on Kerr black hole backgrounds
• Some Remarks on Strichartz Estimates for Homogeneous Wave Equation
• Generalized and weighted Strichartz estimates
• Almost global existence for some semilinear wave equations with almost critical regularity
• Space-time $L^2$ estimates, regularity and almost global existence for elastic waves

Cited by in corpus (3)

• Global existence for some 4-D quasilinear wave equations with low regularity
• Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation
• Global low regularity solutions for nonlinear elastic waves