This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc.

Although there recently have been extensive studies on the perturbation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem on the Vlasov-Maxwell-Boltzmann system with physical angular noncutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a refined time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy X(t)-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo (Invent. Math. 153(3): 593-630 (2003)) to the general collision potentials.

In the paper we are concerned with the large time behavior of solutions to the one-dimensional NavierStokes-Poisson system in the case when the potential function of the self-consistent electric field may take distinct constant states at x = ±∞. Precisely, it is shown that if initial data are close to a constant state with asymptotic values at far fields chosen such that the Riemann problem on the corresponding quasineutral Euler system admits a rarefaction wave whose strength is not necessarily small, then the solution exists for all time and tends to the rarefaction wave as t → +∞. The construction of the nontrivial large-time profile of the potential basing on the quasineutral assumption plays a key role in the stability analysis. The proof is based on the energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.