An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces.

We revisit Yudovich's well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set or on the torus . We construct global-in-time weak solutions with vorticity in and in , where and are suitable uniformly-localized versions of the Lebesgue space and of the Yudovich space respectively, with no condition at infinity for the growth function  . We also provide an explicit modulus of continuity for the velocity depending on the growth function  .

Weak and parabolic solutions of advection-diffusion equations with rough velocity field.

We study the Cauchy problem for the advection-diffusion equation associated with a merely integrable divergence-free vector field defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs.

Anomalous Dissipation and Lack of Selection in the Obukhov-Corrsin Theory of Scalar Turbulence.

The Obukhov-Corrsin theory of scalar turbulence [21, 54] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov's K41 theory of fully developed turbulence [47]. The scaling analysis of Obukhov and Corrsin from 1949 to 1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation.

Growth of Sobolev norms and loss of regularity in transport equations.

We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data [Formula: see text], [Formula: see text], we construct a divergence-free advecting velocity field [Formula: see text] (depending on [Formula: see text]) for which the unique weak solution to the transport equation does not belong to [Formula: see text] for any positive time. The velocity field [Formula: see text] is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space [Formula: see text] that does not embed into the Lipschitz class.

SEM, EBSD, laser confocal microscopy and FE-SEM data from modern shell layers.

Here, we provide the dataset associated with the research article "Orientation patterns of aragonitic crossed-lamellar, fibrous prismatic and myostracal microstructures of modern shells" [1]. Based on several tools (SEM, EBSD, laser confocal microscopy and FE-SEM) we present original data relative to the microstructure and texture of aragonite crystallites in all shell layers (crossed-lamellar, complex crossed-lamellar, fibrous prismatic and pedal retractor and adductor myostraca) and address texture characteristics at the transition from one layer to the other, identifying similarities and differences among the different layers. Shells were cut transversely, obliquely and longitudinally in order to obtain different orientated sections of the outer and inner layer and of the myostraca.