Diffusion limited mixing rates in passive scalar advection
Abstract: 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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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.
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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