Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.     

Reducing Huge Gyroscopic Eigenproblems by Automated Multi-level Substructuring

Simulating numerically the sound radiation of a rolling tire requires the solution of a very large and sparse gyroscopic eigenvalue problem. Taking advantage of the automated multi-level substructuring (AMLS) method it can be projected to a much smaller gyroscopic problem, the solution of which however is still quite costly since the eigenmodes are non-real and complex arithmetic is necessary. This paper discusses the application of AMLS to huge gyroscopic problems and the numerical solution of the AMLS reduction. A numerical example demonstrates the efficiency of AMLS.     

Surfaces of Constant Mean Curvature Bounded by Two Planar Curves

In this paper we study constant mean curvature compact surfaces with two Jordan curves in parallel planes as boundary and we investigate the point at which the surface inherits the symmetries of its boundary.     

Stability of chirped bright and dark soliton-like solutions of the cubic complex Ginzburg-Landau equation with variable coefficients

We consider an inhomogeneous optical fiber system described by the generalized cubic complex Ginzburg-Landau (CGL) equation with varying dispersion, nonlinearity, gain (loss), nonlinear gain (absorption) and the effect of spectral limitation. Exact chirped bright and dark soliton-like solutions of the CGL equation were found by using a suitable ansatz. Furthermore, we analyze the features of the solitons and consider the problem of stability of these soliton-like solutions under finite initial perturbations. It is shown by extensive numerical simulations that both bright and dark soliton-like solutions are stable in an inhomogeneous fiber system. Finally, the interaction between two chirped bright and dark soliton-like pulses is investigated numerically.