Singular integrals and a problem on mixing flows

Hadžić, Mahir; Seeger, Andreas; Smart, Charles K.; Street, Brian

doi:10.1016/j.anihpc.2017.09.001

Cited by 11 publications

(3 citation statements)

References 19 publications

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“…To describe our results, we begin by introducing the following functional (this is different from, but somewhat analogous to, the functionals discussed in [5], [2], [12], [8])…”

Section: Introductionmentioning

confidence: 99%

A new approach to bounds on mixing

Léger

2018

Math. Models Methods Appl. Sci.

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We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L 1 norm of the velocity field. Existing results in the literature use an L p norm with p > 1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.

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“…To describe our results, we begin by introducing the following functional (this is different from, but somewhat analogous to, the functionals discussed in [5], [2], [12], [8])…”

Section: Introductionmentioning

confidence: 99%

A new approach to bounds on mixing

Léger

2018

Math. Models Methods Appl. Sci.

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“…A quantitative Lusin-Lipschitz regularity results for the flow X associated to a vector field b implies lower bounds on the mixing scale of passive scalars driven by b through the transport equation (1.1) (see [56]). In particular, extending the result by Crippa and De Lellis to the case p = 1 would give a positive answer to the well-known Bressan's mixing conjecture proposed in [28] (see also [5][6][7]15,29,30,35,36,41,43,47,50,51,59,62] for related results on both transport and advection-diffusion equations).…”

Section: X(t Y + εR ) − X(t Y)mentioning

confidence: 87%

Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

Nitti

2022

Arch Rational Mech Anal

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We study the regularity of the flow $${\varvec{X}}(t,y)$$ X ( t , y ) , which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution $$\rho \in L^\infty ({\mathbb {R}}^{d+1})$$ ρ ∈ L ∞ ( R d + 1 ) of the continuity equation $$\begin{aligned} \partial _t \rho + {{\,\mathrm{div}\,}}(\rho {\varvec{b}}) = 0, \end{aligned}$$ ∂ t ρ + div ( ρ b ) = 0 , with $${\varvec{b}}\in L^1_t {{\,\mathrm{BV}\,}}_x$$ b ∈ L t 1 BV x . We prove that $${\varvec{X}}$$ X is differentiable in measure in the sense of Ambrosio–Malý, that is $$\begin{aligned} \frac{{\varvec{X}}(t,y+rz) - {\varvec{X}}(t,y)}{r} \underset{r \rightarrow 0}{\rightarrow } W(t,y) z \quad \text {in measure}, \end{aligned}$$ X ( t , y + r z ) - X ( t , y ) r → r → 0 W ( t , y ) z in measure , where the derivative W(t, y) is a BV function satisfying the ODE $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} W(t, y) = \frac{(D {\varvec{b}})_y(\mathrm{d}t)}{J(t-,y)} W(t-, y), \end{aligned}$$ d d t W ( t , y ) = ( D b ) y ( d t ) J ( t - , y ) W ( t - , y ) , where $$(D{\varvec{b}})_y(\mathrm{d}t)$$ ( D b ) y ( d t ) is the disintegration of the measure $$\int D {\varvec{b}}(t,\cdot ) \, \mathrm {d}t$$ ∫ D b ( t , · ) d t with respect to the partition given by the trajectories $${\varvec{X}}(t, y)$$ X ( t , y ) and the Jacobian J(t, y) solves $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} J(t,y) = ({{\,\mathrm{div}\,}}{\varvec{b}})_y(\mathrm{d}t) = \mathrm {Tr}(D{\varvec{b}})_y(\mathrm{d}t). \end{aligned}$$ d d t J ( t , y ) = ( div b ) y ( d t ) = Tr ( D b ) y ( d t ) . The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a $${{\,\mathrm{BV}\,}}$$ BV vector field.

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“…It is well known that the commutator [A, S] and it generalization are elementary operators in harmonic analysis, which play an important role in the theory of the Cauchy integral along Lipschitz curve in C, the boundary value problem of elliptic equation on non-smooth domain, the Kato square root problem on R and the mixing flow problem (see e.g. [2], [4], [13], [24], [10], [18], [8], [25], [19], [23] for the details).…”

Section: Introductionmentioning

confidence: 99%

Maximal Operator for the Higher Order Calderón Commutator

Lai¹

2019

Can. J. Math.-J. Can. Math.

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In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the L p (R d , w) space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.2010 Mathematics Subject Classification. Primary 42B20, 42B25, Secondary 46B70, 47G30.

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Singular integrals and a problem on mixing flows

Cited by 11 publications

References 19 publications

A new approach to bounds on mixing

A new approach to bounds on mixing

Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

Maximal Operator for the Higher Order Calderón Commutator

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