Global smooth solutions of 2-D quadratic quasilinear wave equations with null conditions in exterior domains

For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data smooth solutions have been established in early references. For the Cauchy problem of 2-D quadratic quasilinear wave equations with null conditions, it has been shown that the small data smooth solutions exist globally. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. In the present paper, we solve this open problem through proving the global existence of small solutions in exterior domains. Our main ingredients include: deriving new precise pointwise estimates for the initial boundary value problem of 2-D linear wave equations in exterior domains; finding appropriate divergence structures of quasilinear wave equations under null conditions; introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity, and establishing some crucial pointwise spacetime decay estimates of solutions and their derivatives.

💡 Research Summary

The paper addresses a long‑standing open problem concerning the global existence of small‑amplitude smooth solutions to quadratic quasilinear wave equations in two spatial dimensions when posed in an exterior domain with homogeneous Dirichlet boundary conditions. While the corresponding Cauchy problem (the whole space) has been settled—global existence follows from the null condition—the presence of a boundary in two dimensions introduces severe analytical difficulties that prevent a straightforward extension of the three‑dimensional techniques.

The authors first observe that the standard Klainerman‑Sideris‑Sogge (KSS) type space‑time estimates, which are crucial in three dimensions, rely on the strong Huygens principle and therefore fail in two dimensions. Moreover, the linear wave equation in two dimensions exhibits only logarithmic decay, which is insufficient to control quadratic nonlinearities. To overcome this, the paper develops refined pointwise estimates for solutions of the linear wave equation with compactly supported source terms in an exterior domain. By employing Littlewood‑Paley decompositions and exploiting geometric properties of the unit circle, the authors obtain estimates of the form

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