Global Well-Posedness and Time-Decay Estimates of the Compressible Navier–Stokes–Korteweg System in Critical Besov Spaces

Abstract

We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the L 2 (R d ) -framework.

The global well-posedness for the compressible fluid model of Korteweg type

“…We know the local wellposedness for thses two cases, but for the global well-posedness, our approach does not work. On this point, we refer [5] and [15].…”

The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effects

“…Charve et al [5] obtained the global existence, Gevrey analytic and algebraic time-decay estimates of strong solutions when the initial data are close to a stable equilibrium state in L pcritical framework. In 2019, N. Chikami and T. Kobayashi [8] established the global existence and algebraic time decay estimates of strong solutions when the initial data are close to a stable equilibrium state.…”

Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

“…Define R + = (0, ∞). By Lemma 3.10 and the operator-valued Fourier multiplier theorem due to Weis [37, Theorem 3.4], we have the maximal regularity for (3.2) as follows 5 . Lemma 3.12.…”

“…These global existence results in the whole space are concerned with the the asymptotic stability of the steady state (ρ, u) = (ρ * , 0) satisfying P (ρ * ) > 0, where ρ * is a positive constant. Recently, Chikami and the first author proved in [5] the asymptotic stability of the above steady state satisfying P (ρ * ) = 0. The situation P (ρ * ) = 0 is also treated in [16,17] for the whole space problem.…”