Sensitivity for Smoluchowski equation

TL;DR

This paper studies how solutions to the Smoluchowski coagulation equation depend smoothly on parameters in the interaction kernel, proving differentiability and characterizing derivatives via a linearized equation.

Contribution

It establishes the C^1 regularity of solutions with respect to parameters and provides a rigorous framework for the linearized sensitivity analysis.

Findings

Solution is a C^1 function of (t,lambda)

Derivative solves a linearized equation

Results hold under sub-linear kernel bounds and moment conditions

Abstract

This article investigates the question of sensitivity of the solutions of Smoluchowski equation on R_+^* with respect to parameters \lambda in the interaction kernel K^lambda. It is proved that the solution is a C^1 function of (t,lambda) with values in a good space of measures under the hypotheses K^{lambda}(x,y) \leq phi(x)phi(y), for some sub-linear function phi, a (4+epsilon)-moment assumption on the initial condition, and that the derivative is a solution, in a suitable sense, of a linearized equation.