Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier–Stokes equations with vacuum

Abstract

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the densitydependent Navier-Stokes equations on the whole space R 2 admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure which are the same as those of the homogeneous case.

Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

“…Remark 1.4 It should be noted that our large time decay rates of the velocity and the pressure in (1.11) are the same as those of the incompressible Navier-Stokes equations [24], hence the orientation has no influence on the large time behaviors of the velocity and the pressure.…”

“…Next, using the structure of the 2D heat flows of harmonic maps, we multiply (3.7) by ∇d∆|∇d| 2 and thus obtain some useful a priori estimates on |∇d||∇ 2 d| L 2 and |∇d||∆∇d| L 2 (see (3.23)), which are crucial in deriving the time-independent estimates on both the L ∞ (0, T ; L 2 )-norm of t 1 2 √ ρu and the L 2 (R 2 × (0, T ))-norm of t 1 2 ∇u (see (3.19)). By the similar arguments as [24], we get the bounds of L ∞ (0, T ; L 1 )-norm of spatial weighted estimates of the density (see (3.30)). This together with Lemma 2.5 and some careful analysis indicates the desired L 1 (0, T ; L ∞ ) bound of the gradient of the velocity ∇u (see (3.36)), which in particular implies the bound on the L ∞ (0, T ; L q )-norm (q > 2) of the gradient of the density.…”

“…To this end, we try to adapt some basic ideas used in [24,25], where the authors investigated the global existence of strong solutions to the Cauchy problem of the 2D nonhomogeneous incompressible Navier-Stokes and MHD equations, respectively. However, compared with [24,25], for the incompressible nematic liquid crystal flows treated here, the strong coupling terms and strong nonlinear terms, such as div(∇d ⊙ ∇d), u · ∇d and |∇d| 2 d, will bring out some new difficulties. To overcome these difficulties mentioned above, some new ideas are needed.…”

“…To deal with the difficulty caused by the lack of the Sobolev inequality, we observe that, in the momentum equations (1.2), the velocity u is always accompanied by ρ. Motivated by [11,24], by introducing a weighted function to the density, as well as a Hardy-type inequality (see Lemma 2.3), the ρ η u L r (r ≥ 2, η > 0) is controlled in terms of √ ρu L 2 and ∇u L 2 (see Lemma 2.5). Then we try to obtain the estimates on the L ∞ (0, T ; L 2 )-norm of ∇u L 2 and ∇ 2 d L 2 .…”

“…Then we try to obtain the estimates on the L ∞ (0, T ; L 2 )-norm of ∇u L 2 and ∇ 2 d L 2 . On the one hand, motivated by [24], we use material derivativesu u t + u · ∇u instead of the usual u t , and use some facts on Hardy and BMO spaces (see Lemma 2.6) to bound the key term |P ||∇u| 2 dx (see the estimates of I 2 of (3.6)). On the other hand, the usual L 2 (R 2 × (0, T ))-norm of ∇d t cannot be directly estimated due to the strong coupled term u · ∇d and the strong nonlinear term |∇d| 2 d. Motivated by [26], multiplying (3.7) by ∆∇d instead of the usual ∇d t , and the nonlinear terms u · ∇d and |∇d| 2 d can be controlled in terms of ∇ 2 d and ∇u (see (3.13)), and we find that the key point to obtain the estimate on the L ∞ (0, T ; L 2 (R 2 ))-norm of ∇u and ∇ 2 d is to bound L 2 (0, T ; L 2 (R 2 ))-norm of ∇ 2 d (see (3.15)).…”