Vanishing viscosity limits in the small noise regime for Burgers turbulence

• Grant number: 19/21324-5
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: February 01, 2020
• End date: January 31, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Pedro Jose Catuogno
• Grantee: André de Oliveira Gomes
• Supervisor: Francesco Russo
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Ecole Nationale Supérieure des Techniques Avancées (ENSTA), France
• Associated to the scholarship: 18/06531-1 - Dynamical Systems Perturbed by Lévy Processes, BP.PD

Abstract

The aim of this research plan is to investigate the simultaneous vanishing limit of noise intensity and kinematic viscosity for the stochastic Burgers equation. It is well known that adding a viscosity term or stochastic forcing to first order partial differential equations (PDEs for short) usually turns the resulting solution more regular (non unique solutions become unique or smoother). Soit is a natural question to ask: what happens when both sources of regularization vanish? For the purpose of exploring diverse ways of expressing the solution to the deterministic equation and a full understanding of its stochastic counterpart, available in the literature, we will focus on the one spatial dimension case. We consider the Burgers equation as being stochastically perturbed by two types of forcing: additive white noise and a transport type of noise. For each type of perturbation we develop a different methodology with which we perform the simultaneous zero noise and zero viscosity limit. For the stochastic partial differential equation (SPDE for short) with additive white noise, we develop a pathwise approach. In this case, we link the SPDE to a stochastic system of characteristic differential equations that attempt to offer a way of measuring the concentration of the stochastic solution process around the shocks of the inviscid limit PDE. Ultimately, it is our aim to give estimates for the averages and probabilities associated to this phenomena. For the second type of perturbation, we develop a distributional approach, linking the Burgers SPDE to a stochastic object constituted by differential equations of mean-field type. Both approaches are complementary at the level of reducing complexity of the model. Hopefully we can extract different quantitative knowledge from the two approaches concerning the inviscid zero noise limit for the equations of Burgers turbulent flows. This work is an attempt towards a first direction of research into the phenomenon of the coupled zero noise/zero viscosity for more complicated models of Fluid Mechanics. (AU)