A new approach to bounds on mixing

Léger, Flavien

doi:10.1142/s0218202518500215

Cited by 19 publications

(22 citation statements)

References 15 publications

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“…Our estimate (12) yields the endpoint bound (47) |∂ t V(θ(t, ·))| ≤ C d θ(t, ·) 2 ∞ Dv(t, ·) H 1 . This inequality can be used to extend other results in [15]. For example one obtains the inequality…”

Section: On Léger's Results For Transport Equationsmentioning

confidence: 67%

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Singular integrals and a problem on mixing flows

Hadžić

Seeger

Smart

et al. 2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.

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Section: On Léger's Results For Transport Equationsmentioning

confidence: 67%

“…In a recent preprint Léger [15] considers solutions θ(t, x) of the initial value problem ∂ t θ + div(vθ) = 0 θ(0, ·) = θ 0 on R d ; here v is a given divergence-free time-dependent vector field v on [0, ∞) × R d . See also [16], [14] for related versions of the mixing problem.…”

Section: On Léger's Results For Transport Equationsmentioning

confidence: 99%

Singular integrals and a problem on mixing flows

Hadžić

Seeger

Smart

et al. 2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…for any N ∈ N. We now use the lower bound on our Kantorovich-Rubinstein distance (24) and the assumption on the initial datum to estimate the left-hand side from below. It holds that…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Notice that an estimate of this type fails to be true if the advecting velocity field is only Sobolev regular, see [22,24,8,10,2] for optimal regularity estimates and examples for loss of regularity.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Estimates on mixing by advection were first derived in [15], at least, on the level of the Lagrangian flow, and then translated to the Eulerian setting in [7,34,21,24]; see also [26] for the case of Lipschitz flows. A typical result of these papers indicates that mixing under the (generalized) enstrophy constraint ∇u L p ≤ 1 cannot proceed faster than exponentially in time,…”

Section: Introductionmentioning

confidence: 99%

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Diffusion limited mixing rates in passive scalar advection

Seis¹

2020

Preprint

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We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the budget for the enstrophy is fixed during the evolution, we provide upper bounds on the exponential rates of mixing and of enhanced dissipation. Our results suggest that there is a crossover from advection dominated to diffusion dominated mixing, and we observe a slow down in the exponential decay rates by a logarithm of the diffusivity.

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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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A new approach to bounds on mixing

Cited by 19 publications

References 15 publications

Singular integrals and a problem on mixing flows

Singular integrals and a problem on mixing flows

Diffusion limited mixing rates in passive scalar advection

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

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