On the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations in critical spaces

2023/12/07 by Hammadi Abidi, Guilong Gui, Abidi, Hammadi +3
Mathematics · #35Q30 #76D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2312.03990

openalex publication_date 2023/12/07 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations \eqref1.2 with initial data in the critical spaces. Precisely, under the assumption that the initial velocity u₀ in L² ∩ B^-1+(2)/(p)_p,1 and the initial density ρ₀ in L^∞ and having a positive lower bound, which satisfies 1-ρ₀^-1∈ B^\frac2λ_λ,2∩ L^∞, for p∈[2,∞[ and λ∈ [1,∞[ with (1)/(2)<(1)/(p)+\frac1λ≤1, the system \eqref1.2 has a global solution. The solution is unique if p=2. With additional assumptions on the initial density in case p>2, we can also prove the uniqueness of such solution. In particular, this result improves the previous work in \citeAG2021 where u₀ belongs to B_2,1⁰ and ρ₀^-1-1 belongs to B_(2)/(ε),1^ε, and we also remove the assumption that the initial density is close enough to a positive constant in \citeDW2023 yet with additional regularities on the initial density here.

Related

• Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density
• Global well-posedness for the 2-D inhomogeneous incompressible Navier-Stokes system with large initial data in critical spaces
• Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
• Global solutions to 2-d inhomogeneous navier-stokes system with general velocity
• Density-dependent incompressible Navier--Stokes equations in critical tent spaces