Anna L. Mazzucato

Besov-Morrey spaces: Function space theory and applications to non-linear PDE

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 Karchs 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 pseudo-differential 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.

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L2-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L2 and Lp-Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently.

Vanishing Viscosity Limits for a Class of Circular Pipe Flows

We consider 3D Navier–Stokes flows with no-slip boundary condition in an infinitely long pipe with circular cross section. The velocity fields we consider are independent of the variable parametrizing the axis of the pipe, and the component of the velocity normal to the axis is arranged to be circularly symmetric, though we impose no such symmetry on the component of velocity parallel to the axis. For such flows we analyze the limit as the viscosity tends to zero, including boundary layer estimates.

On the energy spectrum for weak solutions of the Navier?Stokes equations

We consider the decay at high wavenumbers of the energy spectrum for weak solutions to the three-dimensional forced Navier–Stokes equation in the whole space. We observe that known regularity criteria imply that solutions are regular if the energy density decays at a sufficiently fast rate. This result applies also to a class of solutions with infinite global energy by localizing the Navier–Stokes equation. We consider certain modified Leray backward self-similar solutions, which belong to this class, and show that their energy spectrum decays at the critical rate for regularity. Therefore, this rate of decay is consistent with the appearance of an isolated self-similar singularity.

Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows

We 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)]10.1017/S0022112011000292 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 classi...

Mapping properties of heat kernels, maximal regularity, and semi-linear parabolic equations on noncompact manifolds

Let be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form . In particular, we establish results on the distribution kernels and mapping properties of e-tL and (μ + L)s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the Ws,p-Sobolev spaces on M and E. We also prove that L satisfies maximal Lp–Lq-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂tu + Lu = F(t, x, u, ∇ u).

Approximate solutions to second order parabolic equations I: analytic estimates

We establish a new type of local asymptotic formula for the Green’s function Gt(x,y) of a uniformly parabolic linear operator ∂t−L with nonconstant coefficients using dilations and Taylor expansions at a point z=z(x,y) for a function z with bounded derivatives such that z(x,x)=x∊RN. Our method is based on dilation at z, Dyson, and Taylor series expansions. We use the Baker–Campbell–Hausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighed, Lp-type Sobolev spaces Was,p(RN) that appear in practice.

On Transversely Isotropic Elastic Media with Ellipsoidal Slowness Surfaces

We describe all classes of inhomogeneous, transversely isotropic elastic media for which the sheets associated to each wave mode are ellipsoids. These media have the property that elastic waves in each mode propagate along geodesic segments of certain Riemannian metrics. We study the intersection of the sheets of the slowness surface for these media, and, in view of applications to the analysis of propagation of singularities along rays, we give pointwise conditions that guarantee that the sheet of the slowness surface corresponding to a given wave mode is disjoint from the others. We also investigate the smoothness of the associated polarization vectors as functions of position and direction. We employ coordinate and frame-independent methods, suitable to the study of the dynamic inverse boundary problem in elasticity.

A Proof of Einstein's Effective Viscosity for a Dilute Suspension of Spheres

We present a mathematical proof of Einsteins formula for the effective viscosity of a dilute suspension of rigid neutrally buoyant spheres when the spheres are centered on the vertices of a cubic lattice. We keep the size of the container finite in the dilute limit and consider boundary effects. Einsteins formula is recovered as a first-order asymptotic expansion of the effective viscosity in the volume fraction. To rigorously justify this expansion, we obtain an explicit upper and lower bound on the effective viscosity. A lower bound is found using energy methods reminiscent of the work of Keller et al. An upper bound follows by obtaining an explicit estimate for the tractions, the normal component of the stress on the fluid boundary, in terms of the velocity on the fluid boundary. This estimate, in turn, is established using a boundary integral formulation for the Stokes equation. Our proof admits a generalization to other particle shapes and the inclusion of point forces to model self-propelled parti...

Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half Navier–Stokes and Euler equations, respectively.