[Paper Review] Existence of Solutions of the third term of the Connaughton-Newell Model with a source term

Anh Nguyen Thi Nguyen|arXiv (Cornell University)|Mar 15, 2026

Mathematical Biology Tumor Growth0 citations

TL;DR

The paper proves the existence of a solution to the third operator of the Connaughton-Newell equation with a nontrivial source term under a constant interaction kernel and certain regularity assumptions.

ABSTRACT

The Connaughton-Newell equation is an approximation of three-wave kinetic equations using a fully non-linear coagulation-fragmentation model. This equation consists of three non-linear operators. In this paper, we proved that assuming a constant interaction kernel and a well-behaved source term, the third operator of the Connaughton-Newell equation has a solution.

Motivation & Objective

• Motivate the study by connecting wave turbulence with coagulation-fragmentation models and the Connaughton-Newell approximation.
• Formulate the third term S3 of the Connaughton-Newell equation with a source term and constant kernel.
• Establish conditions under which a solution exists for the resulting equation.
• Provide a rigorous proof of existence via analytical techniques and auxiliary constructs.

Proposed method

• Start from the Connaughton-Newell formulation and isolate the third term with a source term and constant kernel (K3 = 1).
• Rewrite the equation in integrated form to obtain a reduced equation for the total quantity N(t).
• Show that the reduced Riccati-type equation N' = 2N^2 + g(t) has nonnegative solutions, given nonnegative initial data and source.
• Introduce auxiliary quantities and use uniform convergence arguments to prove the total wave action integral converges on [0,T].
• Use the Uniform Limit Theorem and Gronwall-type arguments to extend the local existence to the interval [0,T].
• Conclude that the integrated form M(t) = ∫0^∞ Nω dω coincides with N(t) on [0,T], yielding a solution to the original equation.

Experimental results

Research questions

• RQ1 Under what conditions does the third term S3 with a source term admit a solution?
• RQ2 What regularity and nonnegativity assumptions on the initial data and source ensure existence?
• RQ3 Can the existence be extended from local to global time under the stated assumptions?

Key findings

• Under assumptions (A1)–(A4), the equation with a source term admits at least one solution.
• The total wave action ∫0^∞ Nω dω converges uniformly on [0,T] and equals N(t) on [0,T].
• A Riccati-type reduced equation N' = 2N^2 + g(t) governs the integrated quantity and its nonnegative solutions exist.
• The proof uses auxiliary constructs and generating functions to control growth and apply the Weierstrass M-test.
• The result provides a rigorous existence proof for the third term of the Connaughton-Newell model with a source term.