Schulze, Christian. Inviscid mixing by fluid flows. Optimal stirring with cellular and radial velocity fields. 2021, Doctoral Thesis, University of Basel, Faculty of Science.

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Abstract
We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on $\mathbb{R}^d$. The scalar $\rho$ solves the transport equation $\partial_t\rho(t,x) + u(t,x)\cdot\nabla_x\rho(t,x) =0$ with a divergencefree velocity field $u$. We measure the degree of mixedness of the tracer $\rho$ via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scale. For velocity fields with a "fixed palenstrophy constraint" (i.e. a uniformintime bound on the homogeneous Sobolev seminorm $\dot{W}^{s,p}$, where $s>1$ and $1< p \leq \infty$), it is known that the decay of both mixing scales cannot be faster than exponentially fast. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially and thus this structure slows down the mixing process in a significant way. We further construct an explicit example of an autonomous, smooth velocity field $u$ that solves the Euler equation and mixes polynomially fast.
Advisors:  Crippa, Gianluca and Coti Zelati, Michele 

Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 
UniBasel Contributors:  Crippa, Gianluca 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  14105 
Thesis status:  Complete 
Number of Pages:  101 
Language:  English 
Identification Number: 

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Last Modified:  25 Jun 2021 04:30 
Deposited On:  24 Jun 2021 09:23 
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