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 W s,p , where s ≥ 0 and 1 ≤ p ≤ ∞. 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 normḢ −1 , 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 s = 1 and 1 ≤ p ≤ ∞ (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.

Abstract. This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semilinear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudodifferential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

We consider the mixing behavior of the solutions to the continuity equation associated with a divergence-free velocity field. In this note 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

A new instrument for dynamic helical squeeze flow which superposes oscillatory shear and oscillatory squeeze flow Rev. Sci. Instrum. 83, 085105 (2012) The effects of hydrodynamic interaction and inertia in determining the high-frequency dynamic modulus of a viscoelastic fluid with two-point passive microrheology Phys. Fluids 24, 073103 (2012) MHD free convection flow of a visco-elastic (Kuvshiniski type) dusty gas through a semi infinite plate moving with velocity decreasing exponentially with time and radiative heat transfer AIP Advances 1, 022132 (2011) Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shearthinning behavior Phys. Fluids 22, 101701 (2010) Effects of viscoelasticity on the probability density functions in turbulent channel flowWe consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. ["Optimal stirring strategies for passive scalar mixing," J. Fluid Mech. 675, 465 (2011)] that determines the flow field that instantaneously maximizes the depletion of the H − 1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. ["Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations," SIAM J. Numer. Anal. 50(1), 126-150 (2012)] we establish that the H − 1 mix-norm decays at most exponentially in time if the two-dimensional incompressible flow is constrained to have constant palenstrophy. Finite-time perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations in two dimensions suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H − 1 mixnorm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H − 1 mix-norm. C 2012 American Institute of Physics. [http://dx.

We consider the Cauchy problem for the continuity equation in space dimension d ≥ 2. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces W 1,p , for 1 ≤ p < ∞, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in W r,p , with r > 1, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space W r,p does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential self-similar mixing by incompressible flows (J. Amer. Math. Soc., DOI:https://doi.org/10.1090/jams/913), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (C.