On the strong solutions of the primitive equations in 2D domains (original) (raw)
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A RANS 3D model with unbounded eddy viscosities
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2007
We consider the Reynolds Averaged Navier-Stokes (RANS) model of order one (u, p, k) set in R 3 which couples the Stokes Problem to the equation for the turbulent kinetic energy by k-dependent eddy viscosities in both equations and a quadratic term in the k-equation. We study the case where the velocity and the pressure satisfy periodic boundary conditions while the turbulent kinetic energy is defined on a cell with Dirichlet boundary conditions. The corresponding eddy viscosity in the fluid equation is extended to R 3 by periodicity. Our contribution is to prove that this system has a solution when the eddy viscosities are nondecreasing, smooth, unbounded functions of k, and the eddy viscosity in the fluid equation is a concave function.
2013
Steady-State Flow Flow where both velocity and pressure fields are time-independent. Three-dimensional (or 3D) flow Flow where velocity and pressure fields depend on all three spatial variables. Two-dimensional (or planar, or 2D) flow Flow where velocity and pressure fields depend only on two spatial variables belonging to a portion of a plane, and the component of the velocity orthogonal to that plane is identically zero. Local Solution Solution where velocity and pressure fields are known to exist only for a finite interval of time. Global Solution Solution where velocity and pressure fields exist for all positive times. Regular Solution Solution where velocity and pressure fields satisfy the Navier-Stokes equations and the corresponding initial and boundary conditions in the ordinary sense of differentiation and continuity. At times, we may interchangeably use the words \flow" and \solution". successively re-obtained, by different arguments, by a number of authors including Augustin-Louis Cauchy in 1823, Sim eon Denis Poisson in 1829, Adh emar Jean Claude Barr e de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845. We refer the reader to the beautiful paper by Olivier Darrigol [17], for a detailed and thorough analysis of the history of the Navier-Stokes equations. Even though, for quite some time, their significance in the applications was not fully recognized, the Navier-Stokes equations are, nowadays, at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Geology, Oil Industry, Airplane, Ship and Car Industries, Biology and Medicine. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. Notwithstanding these successes, up to the present time, a number of unresolved basic mathematical questions remain open {mostly, but not only, for the physically relevant case of three-dimensional flow. Undoubtedly, the most celebrated is that of proving or disproving the existence of global 3D regular flow for data of arbitrary \size", no matter how smooth (global regularity problem). Since the beginning of the 20th century, this notorious question has challenged several generations of mathematicians who have not been able to furnish a definite answer. In fact, to date, 3D regular flows are known to exist either for all times but for data of \small size", or for data of \arbitrary size" but for a finite interval of time only. The problem of global regularity has become so intriguing and compelling that, in the year 2000, it was decided to put a generous bounty on it. In fact, properly formulated, it is listed as one of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. However, the Navier-Stokes equations present also other fundamental open questions. For example, it not known whether, in the 3D case, the associated initial-boundary value problem is (in an appropriate function space) well-posed in the sense of Hadamard. Stated differently, in 3D it is open the question of whether solutions to this problem exist for all times, are unique and depend continuously upon the data, without being necessarily \regular". Another famous, unsettled challenge is whether or not the Navier-Stokes equations are able to provide a rigorous model of turbulent phenomena. These phenomena occur when the magnitude of the driving mechanism of the fluid motion becomes sufficiently \large", and, roughly speaking, it consists of flow regimes characterized by chaotic and random property changes for velocity and pressure fields throughout the fluid. They are observed in 3D as well as in two-dimensional (2D) motions (e.g., in flowing soap films). We recall that a 2D motion occurs when the relevant region of flow is contained in a portion of a plane, , and the component of the velocity field orthogonal to is negligible. It is worth emphasizing that, in principle, the answers to the above questions may be unrelated. Actually, in the 2D case, the first two problems have long been solved in the affirmative, while the third one remains still open. Nevertheless, there is hope that proving or disproving the first two problems in 3D will require completely fresh and profound ideas that will open new avenues to the understanding of turbulence. The list of main open problems can not be exhausted without mentioning another outstanding question pertaining the boundary value problem describing steady-state flow. The latter is characterized by timeindependent velocity and pressure fields. In such a case, if the flow region, R, is multiply-connected, it is not known (neither in 2D nor in 3D) if there exists a solution under a given velocity distribution at the boundary of R that merely satisfy the physical requirement of conservation of mass.
arXiv (Cornell University), 2017
In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity v 0 ∈ H 2 (Ω) and the initial temperature T 0 ∈ H 1 (Ω)∩L ∞ (Ω) with ∇ H T 0 ∈ L q (Ω), for some q ∈ (2, ∞). Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to H 2 (Ω). The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the " mismatching " of regularities between the horizontal momentum and temperature equations. To handle this "mismatching" of regularities, we introduce several auxiliary functions, i.e., η, θ, ϕ, and ψ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity v or its vertical derivative ∂ z v. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some L 1 t (W 1,∞ x)type a priori estimates on v, we decompose the velocity into the "temperatureindependent" and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Brézis-Gallouet-Wainger and Beale-Kato-Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work [12], is used, and a new logarithmic type Gronwall inequality is exploited.
Journal de Mathématiques Pures et Appliquées, 2008
In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be nonlocal. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.
Incompressible Navier Stokes Equations
arXiv (Cornell University), 2020
A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /Z 3 has been shown by corresponding author of the present work [1]. Smooth solutions for the z−component momentum equation u z assuming the x and y component equations have vortex smooth solutions have been proven to exist, however the Clay Institute Millennium problem on the Navier Stokes equations was not proven for a general enough vorticity form and [1], [3] and references therein do not prove this as previously thought. The idea was to show that Geometric Algebra can be applied to all three momentum equations by adding any two of the three equations and thus combinatorially producing either u x , u y or u z as smooth solutions at a time. It was shown that using the Gagliardo-Nirenberg and Prékopa-Leindler inequalities together with Debreu's theorem and some auxiliary theorems proven in [1] that there is no finite time blowup for 3D Navier Stokes equations for a constant vorticity in the z direction. In part I of the present work it is shown that using Hardy's inequality for u 2 z term in the Navier Stokes Equations that a resulting PDE emerges which can be coupled to auxiliary pde's which give us wave equations in each of the three principal directions of flow. The present work is extended to all spatial directions of flow for the most general flow conditions. In Part II it is shown for the first time that the full system of 3D Incompressible Navier Stokes equations without the above mentioned coupling consists of non-smooth solutions. In particular if u x , u y satisfy a non-constant zvorticity for 3D vorticity ω, then higher order derivatives blowup in finite time but u z remains regular. So a counterexample of the Navier Stokes equations having smooth solutions is shown. A specific time dependent vorticity is also considered.
Viscous potential free-surface flows in a fluid layer of finite depth
Comptes Rendus Mathematique, 2007
It is shown how to model weakly dissipative free-surface flows using the classical potential flow approach. The Helmholtz-Leray decomposition is applied to the linearized 3D Navier-Stokes equations. The governing equations are treated using Fourier-Laplace transforms. We show how to express the vortical component of the velocity only in terms of the potential and free-surface elevation. A new predominant nonlocal viscous term is derived in the bottom kinematic boundary condition. The resulting formulation is simple and does not involve any correction procedure as in previous viscous potential flow theories . Corresponding long wave model equations are derived.