This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, r − ( p − 1 ) r^{-(p-1)} with p > 2 p>2 , for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameters s ∈ ( 0 , 1 ) s\in (0,1) and γ \gamma satisfying γ > − n \gamma > -n in arbitrary dimensions T n × R n \mathbb {T}^n \times \mathbb {R}^n with n ≥ 2 n\ge 2 . Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann H H -theorem. When γ ≥ − 2 s \gamma \ge -2s , we have exponential time decay to the Maxwellian equilibrium states. When γ > − 2 s \gamma >-2s , our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when γ ≥ − 2 s \gamma \ge -2s , as conjectured by Mouhot and Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.

This paper reports on our effort in modeling realistic astrophysical neutron star binaries in general relativity. We analyze under what conditions the conformally flat quasiequilibrium (CFQE) approach can generate "astrophysically relevant" initial data, by developing an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. We show that the CFQE assumptions significantly violate the Einstein field equations for corotating neutron stars at orbital separations nearly double that of the innermost stable circular orbit (ISCO) separation, thus calling into question the astrophysical relevance of the ISCO determined in the CFQE approach.With the need to start numerical simulations at large orbital separation in mind, we push for stable and long term integrations of the full Einstein equations for the binary neutron star system. We demonstrate the stability of our numerical treatment and analyze the stringent requirements on resolution and size of the computational domain for an accurate simulation of the system.

Residential colleges and universities face unique challenges in providing in-person instruction during the COVID-19 pandemic. Administrators are currently faced with decisions about whether to open during the pandemic and what modifications of their normal operations might be necessary to protect students, faculty and staff. There is little information, however, on what measures are likely to be most effective and whether existing interventions could contain the spread of an outbreak on campus. We develop a full-scale stochastic agent-based model to determine whether in-person instruction could safely continue during the pandemic and evaluate the necessity of various interventions. Simulation results indicate that large scale randomized testing, contact-tracing, and quarantining are important components of a successful strategy for containing campus outbreaks. High test specificity is critical for keeping the size of the quarantine population manageable. Moving the largest classes online is also crucial for controlling both the size of outbreaks and the number of students in quarantine. Increased residential exposure can significantly impact the size of an outbreak, but it is likely more important to control non-residential social exposure among students. Finally, necessarily high quarantine rates even in controlled outbreaks imply significant absenteeism, indicating a need to plan for remote instruction of quarantined students.

We study the dynamical evolution of a large amplitude r-mode by numerical simulations. R-modes in neutron stars are unstable growing modes, driven by gravitational radiation reaction. In these simulations, r-modes of amplitude unity or above are destroyed by a catastrophic decay: A large amplitude r-mode gradually leaks energy into other fluid modes, which in turn act nonlinearly with the r-mode, leading to the onset of the rapid decay. As a result the r-mode suddenly breaks down into a differentially rotating configuration. The catastrophic decay does not appear to be related to shock waves at the star's surface. The limit it imposes on the r-mode amplitude is significantly smaller than that suggested by previous fully nonlinear numerical simulations.

Abstract. In this paper, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R 3 previously proved by Klainerman and Machedon [76], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H α for α > 1. By constructing a specific counterexample, we show that, on T 3 , the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α = 1. This is in contrast to the known results in the non-periodic setting, where the these techniques are known to hold for all α ≥ 1.In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.