A study of the possible interactions between fermions assuming only rotational invariance has revealed 15 forms for the potential involving the fermion spins. We review the experimental constraints on unobserved macroscopic, spin-dependent interactions between electrons in the range below 1 cm. An existing experiment, using 1 kHz mechanical oscillators as test masses, has been used to constrain mass-coupled forces in this range. With suitable modifications, including spin-polarized test masses, this experiment can be used to explore all 15 possible spin-dependent interactions between electrons in this range with unprecedented sensitivity. Samples of ferrimagnetic dysprosium iron garnet have been fabricated in the suitable test mass geometry and shown to have spin densities on the order of 10 20h /cm 3 with very low intrinsic magnetism.
We consider the incompressible inhomogeneous Navier-Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak solutions if the velocity, density, and pressure belong to a range Besov spaces of smoothness 1/3. A density-dependent version of the classical Kármán-Howarth-Monin relation is derived.2010 Mathematics Subject Classification. 76S05,35Q35.
When a Leray-Hopf weak solution to the NSE has a singularity set S of dimension d less than 3-for example, a suitable weak solution-we find a family of new L q L p conditions that guarantee validity of the energy equality. Our conditions surpass the classical Lions-Ladyženskaja L 4 L 4 result in the case d < 1. Additionally, we establish energy equality in certain cases of Type-I blowup. The results are also extended to the NSE with fractional power of the Laplacian below 1.
The goal of this note is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution.The key quantity that controls density distortion is the entropy H = ρ log ρ dx, and the measure of deviation from uniformity is given by a well-known conserved quantity e = u ′ + L ψ ρ, where u is velocity and L ψ is the communication operator with kernel ψ. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study. Date: August 15, 2019. 1991 Mathematics Subject Classification. 92D25, 35Q35, 76N10.
The potential failure of energy equality for a solution u of the Euler or Navier-Stokes equations can be quantified using a so-called 'energy measure': the weak- * limit of the measures |u(t)| 2 dx as t approaches the first possible blowup time. We show that membership of u in certain (weak or strong) L q L p classes gives a uniform lower bound on the lower local dimension of E; more precisely, it implies uniform boundedness of a certain upper s-density of E. We also define and give lower bounds on the 'concentration dimension' associated to E, which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of E measure the departure from energy equality. As an application of our estimates, we prove that any solution to the 3-dimensional Navier-Stokes Equations which is Type-I in time must satisfy the energy equality at the first blowup time.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.