[Paper Review] Two dimensional wave--Klein-Gordon equations with semilinear nonlinearities

Shijie Dong, Zoe Wyatt|arXiv (Cornell University)|Nov 24, 2020

Advanced Mathematical Physics Problems50 references4 citations

TL;DR

This paper establishes small data global existence and pointwise decay for two-dimensional coupled wave–Klein-Gordon systems with critical (in time decay) quadratic nonlinearity in the wave equation and below-critical nonlinearity in the Klein-Gordon equation. Using a vector-field method on hyperboloidal foliations, the authors prove that despite the Klein-Gordon field losing linear behavior near the light cone due to the below-critical nonlinearity, optimal time decay and global existence are preserved under small, compactly supported initial data in high-regularity Sobolev norms.

ABSTRACT

From the work on the weak-null condition by Lindblad and Rodnianski, it is well-known that `bad' quadratic sourcing terms are allowed to appear in coupled semilinear wave equations in three spatial dimensions, provided that such terms appear as sources for `good' variables and that the good variables feed back into the system via `good' sourcing terms. Motivated by these ideas, in this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave--Klein-Gordon equations in two spatial dimensions. In particular, we consider critical semilinear nonlinearities for the wave equation and below-critical semilinear nonlinearities for the Klein-Gordon equation. An interesting feature of our two systems is that if the nonlinearities of our PDEs were to be swapped, the nonlinear term in the wave equation would lead to finite-time blow-up.

Motivation & Objective

• To investigate small data global existence and pointwise decay for coupled wave–Klein-Gordon systems in two spatial dimensions with mixed nonlinearity types.
• To analyze the interplay between critical (in time decay) and below-critical nonlinearities in the wave and Klein-Gordon equations, respectively.
• To extend the understanding of nonlinear wave–Klein-Gordon interactions in low dimensions, particularly where linear decay is slow and null-structure is absent.
• To develop and apply a refined vector-field method adapted to hyperboloidal foliations in Minkowski spacetime for 2D systems.
• To provide a foundational framework for studying coupled massive and massless field interactions in mathematical physics.

Proposed method

• Adaptation of the vector-field method to a hyperboloidal foliation of the forward light cone in two-dimensional Minkowski spacetime.
• Use of weighted energy norms and pointwise decay estimates for vector fields applied to both wave and Klein-Gordon components.
• Application of Sobolev embedding and Hardy-type inequalities on hyperboloids to control pointwise growth of derivatives.
• Introduction of a refined bootstrap argument using energy and pointwise decay estimates for both fields simultaneously.
• Decomposition of nonlinear terms into dyadic frequency and angular regions to exploit decay and structure in the nonlinearity.
• Use of the null condition's absence in the wave equation, compensated by the below-critical nature of the Klein-Gordon nonlinearity.

Experimental results

Research questions

• RQ1 Can small-data global existence be established for coupled wave–Klein-Gordon systems in two spatial dimensions when the wave nonlinearity is critical and the Klein-Gordon nonlinearity is below-critical?
• RQ2 How does the presence of a below-critical nonlinearity affect the asymptotic behavior of the Klein-Gordon field near the light cone, despite optimal time decay?
• RQ3 What modifications to the standard vector-field method are required to handle the lack of scaling symmetry and the slow decay in two dimensions?
• RQ4 Can the interplay between critical and subcritical nonlinearities lead to global existence even without the null condition?
• RQ5 What role does the hyperboloidal foliation play in controlling the long-time behavior of solutions in low dimensions?

Key findings

• Global existence of small-data solutions is established for both Model I and Model II systems in two spatial dimensions.
• The Klein-Gordon field exhibits optimal time decay despite losing linear behavior near the light cone due to the below-critical nonlinearity.
• The wave field satisfies critical decay behavior, consistent with the absence of the null condition in the nonlinearity.
• A refined bootstrap argument closes the energy and pointwise decay estimates, proving that the solution exists globally in time.
• The energy and pointwise norms grow at most like $ C_1 ho^ ho $ for some $ \delta > 0 $, with $ \rho = s $, ensuring integrability and decay.
• The method successfully handles the lack of scaling invariance and slow decay in 2D by combining hyperboloidal energy estimates with sharp pointwise control.