Green Functions for the Pressure of Stokes Systems

We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $\Omega \subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions and $\Omega $ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $\Omega $ has a $C^{1,\textrm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considered.

The Coarse Scale Velocity

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

Terms Involving the Divergence Equation

This chapter estimates the terms in the stress which involve solving a divergence equation of the form ∂ⱼQsuperscript jl = Usuperscript l = esuperscript iGreek Small Letter Lamda Greek Small Letter Xiusuperscript l. These terms are the High–Low Interaction term, the main High–High terms, the remainder of the High–High terms, and the Transport term. For each of these factors, the parametrix expansion for the divergence equation is used. The error of the expansion is eliminated by solving the divergence equation. The chapter also considers the bounds which are obeyed for the parametrices of the oscillatory terms and concludes by applying the parametrix.

The Divergence Equation

This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a smooth function and a smooth vector field on ³, plus an unknown, symmetric (2, 0) tensor. The chapter presents a proposition that takes into account a condition relating to the conservation of momentum as well as a condition that reflects Newton's law, which states that every action must have an equal and opposite reaction. This axiom, in turn, implies the conservation of momentum in classical mechanics. In view of Noether's theorem, the constant vector fields which act as Galilean symmetries of the Euler equation are responsible for the conservation of momentum. The chapter shows proof that all solutions to the Euler-Reynolds equations conserve momentum.

Stress Terms Not Involving Solving the Divergence Equation

This chapter estimates the terms in the new stress that do not involve solving the divergence equation. These terms are the Mollification terms and the Stress term. Throughout the estimates, Bsubscript Greek Small Letter Lamda will be assumed to be some constant. After considering the Mollification term from the velocity, the chapter introduces a proposition stating that for k = 0, … , L, there exist constants Cₖ depending on Bsubscript Greek Small Letter Lamda. It then estimates the material derivative, highlighting wastefulness in the estimate, and discusses a commutator estimate suggesting that it may be important to work with frequency energy levels of order L greater than or equal to 2. Finally, it presents the Mollification term from the stress as well as estimates for the Stress term.

Constructing Continuous Solutions

This chapter demonstrates how the preceding construction, combined with a few estimates from Part V, can be used to prove the Main Lemma for continuous solutions. The first step is to mollify the velocity, followed by mollification of the stress. The lifespan is then chosen, preferring a small parameter to ensure that the first term in the parametrix for the High–High term is controlled. The chapter proceeds by discussing the bounds for the new stress and solving the divergence equation, along with the bounds for the corrections and finally, control of the energy increment. The equation for the energy increment includes a smooth vector field and involves bounding the error term.

Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.

Three dimensional simulations and stability analysis for convection induced by absorption of radiation

Purpose – The purpose of this paper is to investigate a model for convection induced by the selective absorption of radiation in a fluid layer. The concentration based internal heat source is modelled quadratically. Both linear instability and global nonlinear energy stability analyses are tested using three dimensional simulations. The results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold. Design/methodology/approach – The author introduce the stability analysis of the problem of convection induced by absorption of radiation in fluid layer, then the author select a situations which have very big subcritical region. Then, the author develop a three dimensions simulation for the problem. To do this, first, the author transform the problem to velocity – vorticity formulation, then the author use a second order finite difference schemes. The author use implicit and explicit schemes to enforce the free divergence equation. The size of the Box is evaluated according to the normal modes representation. Moreover, the author adopt the periodic boundary conditions for velocity and temperature in the $x, y$ dimensions. Findings – This paper explores a model for convection induced by the selective absorption of radiation in a fluid layer. The results demonstrate that the linear instability thresholds accurately predict the onset of instability. A three-dimensional numerical approach is adopted. Originality/value – As the author believe, this paper is one of the first studies which deal with study of stability of convection using a three dimensional simulation. When the difference between the linear and nonlinear thresholds is very large, the comparison between these thresholds is very interesting and useful.