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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

This paper develops L p - L q decay estimates of the gradient of the Stokes semigroup (T (t)) t ≥ 0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω ⊂ R n , n ≥ 3. More precisely, the L p - L q estimates of ∇ T (t) with optimal rates are proved if p and q satisfy | 1 / p − 1 / 2 | < 1 / (2 n) + ε , | 1 / q − 1 / 2 | < 1 / (2 n) + ε , and p ≤ q ≤ n with some ε > 0 , which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L ∞ (0 , ∞ ; L σ 3 (Ω)) provided that the initial velocity is small in the L 3 -norm. [ABSTRACT FROM AUTHOR]

We show that the Stokes operator defined on L σ p (Ω) for an exterior Lipschitz domain Ω ⊂ R n (n ≥ 3) admits maximal regularity provided that p satisfies | 1 / p − 1 / 2 | < 1 / (2 n) + ε for some ε > 0. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on L σ p (Ω) for such p. In addition, L p - L q -mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions to the Navier–Stokes equations in the critical space L ∞ (0 , T ; L σ 3 (Ω)) (locally in time and globally in time for small initial data). [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]

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]