110 research outputs found
Exponential self-similar mixing and loss of regularity for continuity equations
We consider the mixing behaviour of the solutions of the continuity equation
associated with a divergence-free velocity field. In this announcement we
sketch two explicit examples of exponential decay of the mixing scale of the
solution, in case of Sobolev velocity fields, thus showing the optimality of
known lower bounds. We also describe how to use such examples to construct
solutions to the continuity equation with Sobolev but non-Lipschitz velocity
field exhibiting instantaneous loss of any fractional Sobolev regularity.Comment: 8 pages, 3 figures, statement of Theorem 11 slightly revise
Exponential self-similar mixing by incompressible flows
We study the problem of the optimal mixing of a passive scalar under the
action of an incompressible flow in two space dimensions. The scalar solves the
continuity equation with a divergence-free velocity field, which satisfies a
bound in the Sobolev space , where and . The mixing properties are given in terms of a characteristic length
scale, called the mixing scale. We consider two notions of mixing scale, one
functional, expressed in terms of the homogeneous Sobolev norm ,
the other geometric, related to rearrangements of sets. We study rates of decay
in time of both scales under self-similar mixing. For the case and (including the case of Lipschitz continuous velocities, and
the case of physical interest of enstrophy-constrained flows), we present
examples of velocity fields and initial configurations for the scalar that
saturate the exponential lower bound, established in previous works, on the
time decay of both scales. We also present several consequences for the
geometry of regular Lagrangian flows associated to Sobolev velocity fields.Comment: To appear in Journal of the American Mathematical Society. Some
results were announced in G. Alberti, G. Crippa, A. L. Mazzucato,
"Exponential self-similar mixing and loss of regularity for continuity
equations", C. R. Math. Acad. Sci. Paris, 352(11):901--906, 2014,
arXiv:1407.2631v
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