Nonstationary Stokes equations in a half-space with continuous initial data

The paper is devoted to nonstationary Stokes equations in a half-space. The existence and uniqueness of a solution are proved in spaces of bounded or continuous functions. Estimates of solutions are given in the uniform norm and in the norms of Hölder spaces. Bibliography: 17 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 118–167.     

A Remark on the Stokes Problem with Initial Data in L 1

In this note we study the initial boundary value problem of the Stokes system. We assume the initial data belonging to the Lebesgue space L ¹. We develop a suitable approach to discuss the existence, uniqueness and continuous dependence on the data. As matter of course, we also consider the case of the Lebesgue space L ^p , p ? (1,￥){p\in(1,\infty)}.     

Navier–Stokes equations in aperture domains: Global existence with bounded flux and time‐periodic solutions

We consider the Navier–Stokes equations in an aperture domain of the three‐dimensional Euclidean space. We are interested in proving the existence of regular solutions corresponding to small initial data and flux through the aperture. The flux is assumed to be smooth and bounded on (0, +∞). As a consequence, we prove the existence of a time‐periodic solution corresponding to a time‐periodic flux through the aperture. Finally, we compare our solution with a solution belonging to a wider class, showing that, if such a solution does exist, then the two solutions coincide. Copyright © 2007 John Wiley & Sons, Ltd.     

Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains

For an arbitrary domain Ω ⊂ ℝⁿ, n=2,3, Ω ≠ ℝⁿ, we prove the existence of weak periodic solutions to the Navier-Stokes equations and of regular solutions if the data are small or satisfy certain symmetry conditions. We also show that the periodic regular solutions are stable. Bibliography: 38 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 142–182.     

“Pointwise Asymptotic Stability of Steady Fluid Motions”

We study pointwise asymptotic stability of steady incompressible viscous fluids. The region of the motion is bounded. Our results of stability are based on the maximum modulus theorem that we prove for solutions of the Navier–Stokes equations. The asymptotic stability is based on a variational formulation. Since the region of the motion is bounded, the time decay is of exponential type. Of course suitable assumptions are made about the smallness of the size of the uniform norm of the perturbations at the initial data. With no restrictions, we are able only to prove an existence theorem of the perturbation local in time.     

Navier–Stokes equations: an analysis of a possible gap to achieve the energy equality

The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is to evaluate the possible gap between the energy equality and the energy inequality deduced for a weak solution. This kind of analysis is new and the result is a natural continuation and improvement of a result obtained by the same authors in Crispo et al. (Some new properties of a suitable weak solution to the Navier–Stokes equations. arXiv:1904.07641).

     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.