[Paper Review] Stability of a wave and Klein-Gordon system with mixed coupling

Shijie Dong|arXiv (Cornell University)|Dec 11, 2019

Advanced Mathematical Physics Problems29 references4 citations

TL;DR

This paper establishes global existence and sharp pointwise decay for a coupled system of semilinear wave and Klein-Gordon equations with mixed quadratic nonlinearities, despite the absence of derivatives on the wave component in nonlinear terms. By introducing a novel transformation to reveal a hidden null structure and employing the scaling vector field on the wave component, the author derives improved $L^2$-type estimates and closes the bootstrap argument via distinguished high- and low-order energy estimates, proving small data global existence without assuming null conditions.

ABSTRACT

We are interested in establishing stability results for a system of semilinear wave and Klein-Gordon equations with mixed coupling nonlinearities, that is, we consider all of the possible quadratic nonlinear terms of the type of wave and Klein-Gordon interactions. The main difficulties are due to the absence of derivatives on the wave component in the nonlinearities. By doing a transformation on the wave equation, we reveal a hidden null structure. Next by using the scaling vector field on the wave component only, which was generally avoided, we are able to get very good $L^2$--type estimates on the wave component. Then we distinguish high order and low order energies of both wave and Klein-Gordon components, which allows us to close the bootstrap argument.

Motivation & Objective

• To establish small data global existence for a coupled system of semilinear wave and Klein-Gordon equations with all possible quadratic nonlinearities.
• To resolve the challenge posed by nonlinear terms lacking derivatives on the wave component, which lead to critical $L^2$-norm decay of $t^{-1}$.
• To overcome the failure of standard energy methods due to the absence of derivative control on the wave component in nonlinearities.
• To extend previous results by handling nonlinearities of the type $u\partial v$ in the Klein-Gordon equation, which were not treated in earlier works.
• To achieve sharp pointwise decay estimates for both wave and Klein-Gordon components under minimal structural assumptions.

Proposed method

• Introduce a transformation on the wave equation to reveal a hidden null structure in the nonlinearities.
• Apply the scaling vector field to the wave component, a technique generally avoided, to derive strong $L^2$-type estimates.
• Distinguish high-order and low-order energies for both wave and Klein-Gordon components to manage the growth in nonlinear interactions.
• Use the hyperboloidal foliation method to derive energy estimates on spacelike hypersurfaces and control pointwise decay.
• Employ commutator estimates and conformal-type energy estimates to handle the interaction between wave and Klein-Gordon components.
• Close the bootstrap argument by combining refined $L^rown{\infty}$, $L^2_f$, and $L^2$ estimates on hyperboloids, ensuring integrability of nonlinear terms.

Experimental results

Research questions

• RQ1 Can global existence be established for a coupled wave-Klein-Gordon system with all possible quadratic nonlinearities, including those with no derivatives on the wave component?
• RQ2 Does the absence of derivatives on the wave component in nonlinear terms prevent global existence, or can a hidden null structure compensate for this?
• RQ3 Can the scaling vector field be effectively used on the wave component to derive $L^2$-type estimates in a system with mixed coupling?
• RQ4 Is it possible to achieve sharp pointwise decay for both components without assuming the null condition on the nonlinearities?
• RQ5 How can high- and low-order energy estimates be combined to close the bootstrap argument in a system with critical non-integrable nonlinearities?

Key findings

• The system admits small data global solutions for all quadratic nonlinearities of wave-Klein-Gordon type, even without assuming the null condition.
• The nonlinearities $Q_{u0}(u;v,\partial v)$ and $Q_{v0}(u;v,\partial v)$, which include terms like $uv$ and $u\partial v$, are shown to be integrable in time due to improved decay estimates.
• The $L^2$-type estimates for the wave component are significantly enhanced by applying the scaling vector field, enabling control of the critical $t^{-1}$ decay behavior.
• Sharp pointwise decay rates of $t^{-3/2}$ for the wave component and $t^{-1}$ for the Klein-Gordon component are established, consistent with the expected optimal decay for such systems.
• The refined energy estimates close the bootstrap argument, proving that the solution exists globally in time for sufficiently small initial data.
• The method successfully handles nonlinear terms of the form $u\partial v$ in the Klein-Gordon equation, extending prior results that excluded such terms.