Global solutions to isentropic compressible Navier-Stokes equations in 3D thin domains

In this paper, we study the global well-posedness of isentropic compressible Navier-Stokes equations in three-dimensional (3D) periodic thin domains of type $$\mathbb{T}\times(\delta\mathbb{T})^{2}$$ , where 0 < δ < 1 is a small parameter. We apply Littlewood-Paley decomposition theory to the periodic thin domain $$\mathbb{T}\times(\delta\mathbb{T})^{2}$$ and show some Bernstein type inequalities with specific dependence on the parameter δ. This allows us to establish various embedding inequalities in Besov spaces in $$\mathbb{T}\times(\delta\mathbb{T})^{2}$$ as well as the interpolation inequalities of Gagliardo-Nirenberg type. Together with ideas of Hoff (1995), we prove that the compressible Navier-Stokes equations in $$\mathbb{T}\times(\delta\mathbb{T})^{2}$$ admit a unique global regular solution when the thickness δ of the domain is sufficiently small, even if the initial data (ρ0,δ, u0,δ) are large in the sense that $$\Vert(\nabla^{2}\rho_{0},\nabla^{2}u_{0,\delta})\Vert_{L^{2}(\mathbb{T}\times(\delta\mathbb{T})^{2})}\sim\delta^{-\kappa}$$ with $$\kappa\in(0,{1\over{2}})$$ .