The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].

We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds.

We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.

We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for a one-dimensional (1D) bipolar Vlasov--Poisson--Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type of decomposition around the local Maxwellian related to the bipolar VPB system in the previous work [H.-L. Li, Y. Wang, T. Yang, and M.-Y. Zhong, <i>Arch. Ration. Mech. Anal.</i>, 228 (2018), pp. 39--127], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to the 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical superposition wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential

In this note, we study submanifold geometry of the Atiyah--Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over S^2, the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the ``strong stability condition" proposed in our earlier work, and confirm the global uniqueness as a corollary.