Global Solutions of a General Hyperbolic Navier-Stokes Equations

Abstract

Whether or not classical solutions to the hyperbolic Navier-Stokes equations (NSE) can develop finite-time singularities remains a challenging open problem. For general data without smallness condition, even the L²-norm of solutions is not known to be globally bounded in time. This paper presents a systematic approach to the global existence and stability problem by examining the difference between a general hyperbolic NSE and its corresponding Navier-Stokes counterpart. We make use of the integral representations. The functional setting is taken to be critical Sobolev spaces for the NSE. As a special consequence, any d-dimensional (d ≥ 2) hyperbolic NSE with general fractional dissipation is shown to possess a unique global solution if the coefficient of the double-time derivative and the initial data obey a suitable constraint.

Keywords

Critical Sobolev space / global well-posedness / hyperbolic Navier-Stokes equations / 35A05 / 35Q35 / 76D03

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Peking University