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.
The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L 2 (R) maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy f 0 L ∞ < ∞ and ∂ x f 0 L ∞ < 1. We take advantage of the fact that the bound ∂ x f 0 L ∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance f 1 ≤ 1/5. Previous results of this sort used a small constant ǫ ≪ 1 which was not explicit [5,16,7,12].
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in R 3 with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in [11,1] that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound |v(x, t)| ≤ C * (r 2 − t) −1/2 for −T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large, we then prove that v is regular at time zero.
The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over R 3 . It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is 1/4. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.
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.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.