On the Concentrations for Isentropic Compressible Navier-Stokes Equations

Abstract

This thesis is devoted to the study from the mathematical aspects on the dynamics of viscous compressible fluids, more specifically, the existence of global-in-time weak solutions to the isentropic compressible Navier-Stokes equations, which follows from the compactness of the solution set in the weak topology. We shall mainly focus on the open part of this problem, namely, the existence when the adiabatic exponent 𝛾≥1 does not exceed half of the space dimension 𝑁, and one of the most severe obstructions–the development of concentrations in the limiting process–will be extremely addressed in this dissertation.

First of all, using Federer’s structure theorem we establish a concentration cancellation result, namely, that when a concentration set has locally finite one-dimensional Hausdorff measure, the weak limit of a solution sequence remains a weak solution. Moreover, by verifying that a system of divergence-free equations has only trivial solutions, it is proved that any solution sequence with uniformly bounded energy either converges strongly in the corresponding energy spaces or admits the weak limit that is not a weak solution.

In the second part, the size of the possible concentration set is estimated. It is proved that no concentration occurs except for a space-time set of cylindrical Hausdorff dimension 𝑁−2𝑁𝛾/(𝑁+𝛾(𝑁−2𝛾)). Besides, by developing a criterion for the elimination of concentations in terms of the local decay of the gradient of the velocity field, we deduce from a rather standard covering argument that the concentration can only occur within a space-time set of parabolic Hausdorff dimension 𝑁−2+max{0,min(2/𝛾,(𝑁+6)/𝛾−2𝛾)} under a reasonable assumption. It is remarkable that both the above dimensions are one, and thus, almost optimal for the critical case 𝛾=𝑁/2 in view of our concentration cancellation theorem.

Date of Award	22 Jul 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Panpan REN (Supervisor) & Xianpeng HU (External Co-Supervisor)