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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

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]

We consider a free boundary problem of the Navier–Stokes equations in the three-dimensional Euclidean space with moving contact line, where the 90 ∘ -contact angle condition is posed. We show that for given T > 0 the problem is local well-posed on (0 , T) provided that the initial data are small. We study the transformed problem in an L p -in-time and L q -in-space setting with 2 < p < ∞ and 3 < q < ∞ satisfying 2 / p + 3 / q < 1 , which provides a weaker setting than the one that was allowed by Wilke (2020) whose case was p = q > 5. [ABSTRACT FROM AUTHOR]

Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains Ω t + , Ω t − ⊂ R N , N ≥ 2 , where the domains are separated by a sharp compact interface Γ t ⊂ R N − 1 . We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are sufficiently small and the initial domain of the incompressible fluid is close to a ball. In particular, we obtain the solution in the maximal L p − L q -regularity class with 2 < p < ∞ and N < q < ∞ and exponential stability of the corresponding analytic semigroup on the infinite time interval. [ABSTRACT FROM AUTHOR]

We consider a free boundary problem of compressible–incompressible two-phase flows with phase transitions in general domains of N -dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The compressible fluid and the incompressible fluid are separated by either compact or non-compact sharp moving interface, and the surface tension is taken into account. In our model, the compressible fluid and incompressible fluid are occupied by the Navier–Stokes–Korteweg equations and the Navier–Stokes equations, respectively. This paper shows that for given T > 0 the problem admits a unique strong solution on (0 , T) in the maximal L p − L q regularity class provided the initial data are small in their natural norms. [ABSTRACT FROM AUTHOR]