Distributional Solutions to Nonlinear Partial Differential Equations

This research paper examines the theory of distributional solutions for nonlinear partial differential equations (PDEs), with a focus on specific equations such as the nonlinear wave equation, nonlinear Schrödinger equation, and equations arising in geometric analysis.The study delves into the existence, uniqueness, and regularity properties of distributional solutions.It begins with an overview of distribution theory and its application to linear PDEs, and then explores how these concepts extend to nonlinear equations.Fundamental aspects such as well-posedness, stability, and convergence of distributional solutions are analyzed, providing rigorous mathematical formulations and proofs.Given the nonlinear PDE N(u)=0,we aim to study its distributional solutions u∈D′(Ω) where Ω⊆ℝ is the domain of interest.Our focus lies on specific instances of N(u)=0, such as the nonlinear wave equation, nonlinear Schrödinger equation, and equations arising in geometric analysis.The investigation begins with an overview of distribution theory and its application to linear PDEs, followed by an extension to nonlinear equations.Key aspects including well-posedness, stability, and convergence of distributional solutions are carefully analyzed, providing rigorous mathematical formulations and proofs.Additionally, the paper discusses implications for various fields including mathematical physics, geometric analysis, and applied mathematics.