Search Results

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.

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.

This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.

This chapter studies the spectrum of Laplace–Beltrami operators on compact manifolds. It begins by defining a metric on an open subset Ω‎ ⊂ Rⁿ, in order to lift their results to corresponding ones on compact manifolds. The chapter then details some elliptic regularity estimates, before embarking on a brief review of geodesics and normal coordinates. The purpose of this review is to show that, with given a particular Laplace–Beltrami operator and any point y₀ in Ω‎, one can choose a natural local coordinate system y = κ‎(x) vanishing at y₀ so that the quadratic form associated with the metric takes a special form. To conclude, the chapter turns to the Hadamard parametrix.

This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ‎) = cλ‎ⁿ + O(λ‎superscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.