On the uniqueness of strong solution to the incompressible Navier–Stokes equations with damping

Abstract

In this paper, we show that the Cauchy problem of the incompressible Navier-Stokes equations with damping α|u| β−1 u(α > 0) has global strong solution for any β > 3 and the strong solution is unique when 3 < β 5. This improves earlier results.

Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping

“…In particular, in [4], the authors have proved the existence of global weak solutions for r ≥ 2, and the existence of global strong solutions for r ≥ 9 2 and the uniqueness of strong solutions for 9 2 ≤ r ≤ 6, respectively, for the three dimensional Navier-Stokes equations with nonlinear damping in the whole space. The results established in [4] have been improved in [31], they proved the existence of global strong solutions for the three dimensional Navier-Stokes equations with nonlinear damping in the whole space with u 0 ∈ H 1 (R 3 ) ∩ L r (R 3 ) under the assumptions that r ≥ 4 and the uniqueness of strong solutions for the three dimensional Navier-Stokes equations with nonlinear damping in the whole space under the assumptions that 4 < r ≤ 6 as well as r = 4 and sufficiently large β > 0. In [33], the author has proved the existence of global strong solutions for the three dimensional Navier-Stokes equations with nonlinear damping in the whole space with u 0 ∈ H 1 (R 3 ) under the assumption that r > 4 as well as r = 4 and sufficiently large β > 0, and also proved that the strong solution is unique in the class of weak solutions for any r ≥ 2, which are significant improvements of those results established in [4].…”

“…However, the uniqueness of weak solutions and the global existence of strong solutions for the three dimensional incompressible Navier-Stokes equations remain open until now. Therefore, many authors turn to consider the well-posedness and the long-time behavior of solutions for problem (1) with β > 0 (see [4,12,13,14,24,25,26,27,31,33]). In particular, in [4], the authors have proved the existence of global weak solutions for r ≥ 2, and the existence of global strong solutions for r ≥ 9 2 and the uniqueness of strong solutions for 9 2 ≤ r ≤ 6, respectively, for the three dimensional Navier-Stokes equations with nonlinear damping in the whole space.…”

On the Weak Solution to a Fractional Nonlinear Schrödinger Equation

“…Interestingly enough, there are also some other models involving the damping term | | ; see [16,17].…”

On a Fractional Nonlinear Hyperbolic Equation Arising from Relative Theory

“…Interestingly enough, the parabolic version of (1)-(3) with convection corresponds to the Navier-Stokes equations with damping; see [13,14].…”