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This article studies the stability of a stationary solution to the three-dimensional Navier-Stokes equations in a bounded domain, where surface tension effects are taken into account. More precisely, this article considers the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in R3{{\mathbb{R}}}^{3}, which are rotationally symmetric about a certain axis. It is proved that this stability result can be obtained by the positivity of the second variation of the energy functional associated with the equation that determines an equilibrium figure, provided that initial data are close to an equilibrium state. The unique global solution is constructed in the Lp{L}^{p}-in-time and Lq{L}^{q}-in-space setting with (p,q)∈(2,∞)×(3,∞)\left(p,q)\in \left(2,\infty )\times \left(3,\infty ) satisfying 2/p+3/q

Let Ω be an exterior Lipschitz domain in R². It is proved that the Helmholtz decomposition of the vector fields in Lp(Ω; R²) exists if p satisfies |1/p−1/2| < 1/4+ε with some constant ε = ε(Ω) ∈ (0, 1/4], where it is allowed to take ε = 1/4 if ∂Ω ∈ C¹. [ABSTRACT FROM AUTHOR]

Consider the chemotaxis–Navier–Stokes equations on a bounded convex domain Ω ⊂ R 3 , where the boundary ∂ Ω of Ω is not necessarily smooth. It is shown that this system admits a unique strong 2 π -periodic solution provided that given 2 π -periodic forcing functions are sufficiently small in their natural norm. The result may extend to general cases Ω ⊂ R d , d ≥ 2 , if one additionally assumes that ∂ Ω is of class C 3. The nonnegativity of solutions is also discussed. [ABSTRACT FROM AUTHOR]