[Paper Review] Existence and regularity of solutions in nonlinear wave equations

Geng Chen, Yannan Shen|arXiv (Cornell University)|Sep 12, 2014

Advanced Mathematical Physics Problems2 references3 citations

TL;DR

This paper establishes the global existence and Hölder regularity of solutions for a family of nonlinear wave equations parameterized by λ ∈ (0, 1/2]. By transforming the equations into a semi-linear system in new coordinates, the authors prove that solutions are Hölder continuous with exponent 1−λ, even when gradient blowup occurs. Numerical evidence confirms that key variables remain bounded, supporting the regularity result for λ ∈ (0, 1/3].

ABSTRACT

In this paper, we study the global existence and regularity of Hölder continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.

Motivation & Objective

• To investigate the dependence of solution regularity on the parameter λ in nonlinear wave equations.
• To establish global existence and Hölder continuity of weak solutions for equations of the form (1.1) and (1.2) with λ ∈ (0, 1/2].
• To provide numerical evidence supporting that solutions remain Hölder continuous with exponent 1−λ even after gradient blowup.
• To bridge the gap in understanding the transition from discontinuous (λ=1) to Hölder continuous (λ=1/2) to smoother (λ→0) solutions.
• To develop a numerical framework based on a semi-linear system to test boundedness of auxiliary variables, which implies Hölder regularity of the original solution.

Proposed method

• Transform the original nonlinear wave equations (1.1) and (1.2) into new coordinates (X,Y) via a change of variables that simplifies the structure of the equations.
• Derive a semi-linear system of equations (3.16) for variables u, v, w, p, q in the (X,Y) plane, which is equivalent to the original system.
• Use iterative integral equations (3.19) to construct solutions in a bounded domain Ω̄_r, leveraging a weighted sup-norm space Λ_r to ensure convergence.
• Apply a contraction mapping argument in the space Λ_r with sufficiently large κ to prove existence of a solution to the semi-linear system.
• Perform numerical experiments on the bounded domain Ω̄_r using initial data u(x,0) = sech(x) and u_t(x,0) = u_x(x,0), with c(u) = √(cos²u + 1).
• Monitor the behavior of auxiliary variables p and q during evolution; their uniform boundedness away from zero and infinity implies Hölder regularity of u with exponent 1−λ.

Experimental results

Research questions

• RQ1 Does the solution of the nonlinear wave equation (1.1) remain Hölder continuous with exponent 1−λ for λ ∈ (0, 1/2] despite gradient blowup?
• RQ2 Can the regularity of solutions to (1.2) be systematically improved as λ decreases from 1 to 0, particularly in the range λ ∈ (0, 1/3]?
• RQ3 Is the boundedness of auxiliary variables p and q in the transformed semi-linear system (3.16) a sufficient condition for the Hölder continuity of the original solution u?
• RQ4 Can numerical experiments on the semi-linear system reliably predict the regularity of solutions to the original quasi-linear wave equation?
• RQ5 How does the transformation to (X,Y) coordinates facilitate the analysis of global existence and regularity in the presence of gradient blowup?

Key findings

• For λ ∈ (0, 1/3] ∪ {1/2}, solutions to equation (1.1) are Hölder continuous in both x and t with exponent 1−λ.
• Numerical experiments for λ = 1/4 and λ = 1/3 show that the auxiliary variables p and q remain uniformly positive and bounded even when w attains π (indicating gradient blowup).
• The convergence of the iterative scheme (3.19) in the weighted space Λ_r with large κ confirms the existence of a solution to the semi-linear system (3.16).
• The boundedness of p and q in the (X,Y) coordinates implies that the original solution u(x,t) is Hölder continuous with exponent 1−λ, even after finite-time gradient blowup.
• The transformation to (X,Y) coordinates and the resulting semi-linear system allow for a more tractable numerical analysis of Hölder regularity compared to direct analysis of the quasi-linear system.
• The results support the conjecture that solution regularity improves as λ decreases, with λ → 0 corresponding to a limit approaching C¹ regularity.