Global Existence of Classical Solutions to Brenner-Navier-Stokes-Fourier System for Large Data

TL;DR

This paper proves the global existence of classical solutions for the 1D Brenner-Navier-Stokes-Fourier system with large initial data, using advanced mathematical techniques to establish bounds and dissipative structures.

Contribution

It establishes the global existence of classical solutions for large initial data in the 1D BNSF system, a refinement of classical fluid models, with novel bounds on specific volume and temperature.

Findings

Proved global existence of solutions for large initial data.

Established bounds for specific volume and temperature.

Applied De Giorgi method and maximum principle for bounds.

Abstract

We study the 1D Brenner-Navier-Stokes-Fourier (BNSF) system, proposed as a refinement of the classical Navier--Stokes--Fourier model through the introduction of the volume velocity, distinct from the mass velocity describing convective transport. When formulated in the Lagrangian mass coordinates with the volume velocity, the discrepancy between the two velocities induces a dissipative structure in the mass conservation law. We prove the global existence of classical solutions for arbitrarily large initial data. More precisely, for initial data in $H^k(\mathbb{R})$ with $k\ge3$, with the specific volume and absolute temperature initially bounded away from zero, we construct global-in-time solutions that remain in the same regularity class. Our result accommodates arbitrarily large initial data. A major difficulty is to establish lower and upper bounds for the specific volume \(v\). The additional dissipation yields an $L_t^2 L_x^2$ bound for $v_x$, which is further improved to an $L_t^\infty L_x^\infty$ bound of $v$ and $1/v$ via the parabolic De Giorgi method. We also apply the maximum principle to obtain a positive lower bound for the absolute temperature.