Strong Solutions to Non-Stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heating

TL;DR

This paper proves the local existence, uniqueness, and smoothness of solutions for non-stationary heat-conducting viscous incompressible fluid flows in channels with complex boundary conditions, over a certain time interval.

Contribution

It establishes the first rigorous results on strong solutions for non-stationary heat-conducting incompressible flows with 'do nothing' boundary conditions.

Findings

Proves local in time existence of solutions.

Demonstrates global uniqueness of solutions.

Shows smoothness of solutions under specified conditions.

Abstract

We study an initial-boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in channel-like domains on a time interval $(0,T)$. For the parabolic system with strong nonlinearities and including the artificial (the so called "do nothing") boundary conditions, we prove the local in time existence, global uniqueness and smoothness of the solution on a time interval $(0,T^*)$, where $0< T^* \leq T$.