Global Kato's Solution for Inhomogeneous Navier-Stokes System in L³(R³)

Abstract

In this thesis, the inhomogeneous incompressible Navier-Stokes system is studied. We consider the global-in-time well-posedness of Cauchy problem for the inhomogeneous incompressible Navier-Stokes system when the initial velocity v₀ is small in the critical space L³ (R³), and when the initial density p₀ with small fluctuation is away from the vacuum. Due to the appearance of the initial layer, the initial velocity v₀ is assumed to be bounded in the critical space Ḣ ^½(R³).

The local estimate of the pressure term is given with the help of the weighted global Ḣ ^½ energy inequality, local L² and H¹ energy inequalities. Moreover, the pressure term is split into two parts to estimate. The first part P₁ takes the contribution purely from the homogeneous incompressible Navier-Stokes system, while the second part P₂ takes the contribution from the appearance of the initial layer. The first part P₁ is estimated by the method proposed in [32] and the second part P₂ relies on the weighted global Ḣ ^½ energy inequality to give a weighted L² energy estimate.

A direct proof of the partial regularity result of the inhomogeneous incompressible Navier-Stokes system is given. We avoid using proof by contradiction, hence the process seems more clear. The argument in [8] is also considered to give a key estimate.

At last, the existence of solutions in L_x³L_t^∞(R³ x R⁺) to the inhomogeneous incompressible Navier-Stokes system is proved. A covering argument, which gives a method to handle the system without L³-monotonicity in time, is applied.

Date of Award	5 Aug 2024
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Shun ZHANG (Supervisor) & Xianpeng HU (Supervisor)