Polynomial growth estimates for multilinear singular integral operators

Abstract. We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. We use x ∈ Ω ⊆ R 3 to denote the material point and f (x, m, t) to represent the number density for the number of molecules whose orientation is parallel to m at point x and time t. where De is the Deborah number, R is the rotational gradient operator(see Section 3), κ is a constant velocity gradient, and U is the mean-field interaction potential. Onsager [18]

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“…whose L 2 norm is bounded by ∇f 3 L ∞ f 2 L 2 ; see [2] for example. Now we are ready to prove Proposition 7.2.…”

Section: This Means Thatmentioning

confidence: 99%

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

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“…Under regularity assumptions analogous to (1.1) or (1.2), the theory has been developed through works of Coifman-Meyer [2], [3], [4], Christ-Journé [6], Kenig-Stein [15] and Grafakos-Torres [10], [11]. Some of the most recent work in the subject was motivated in part by the results of Lacey-Thiele [16], [17] on the bilinear Hilbert transform and a search for the optimal range of exponents where boundedness in Lebesgue spaces can be obtained.…”

Section: A R L O S ṕ Erez and Rodolfo H Torresmentioning

confidence: 99%

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

Pérez

Torres

2014

Proc. Amer. Math. Soc. Ser. B

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“…Christ and Journé [6] studied multilinear forms U , so that the bilinear forms obtained by fixing all but two of their arguments,…”

Section: Examples and Some Previous Work In The Subjectmentioning

confidence: 99%

“…the work of Nazarov, Treil, and Volberg [49]), but in our general situation T is only a priori defined on smooth functions. Using different arguments, Christ and Journé established in [6] another version of the multilinear T 1 Theorem. Consider the (m + 1)-linear form defined on functions in…”

Section: The Multilinear T 1 Theoremmentioning

confidence: 99%

“…We express this result in terms of the action of an operators and its multilinear transposes on the quintessential building blocks of Fourier analysis: the characters x → e 2πix·ξ . A different formulation of a T 1 Theorem for multilinear forms was given before by Christ and Journé [6]. Our version is very well-suited for a couple of applications to pseudodifferential and translation invariant operators that we describe in Corollary 2 and Corollary 3.…”

Section: Introductionmentioning

confidence: 99%

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On multilinear singular integrals of Calderón-Zygmund type

Grafakos

Torres²

2002

Publicacions Matemàtiques

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Abstract. A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T 1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.

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Polynomial growth estimates for multilinear singular integral operators

Cited by 164 publications

References 14 publications

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

On multilinear singular integrals of Calderón-Zygmund type

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