A linear hyperbolic system and an optimal control problem

We prove a theorem of existence, uniqueness, and continuous dependence for a linear hyperbolic system with Darboux-type conditions under assumptions on the coefficients, which are in a sense the most general possible. Moreover, an application of this result to an optimal control problem is given.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

On the spatially homogeneous landau equation for hard potentials part i : existence,uniqueness and smoothness

We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity.     

On the spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications

We find a lower bound for the entropy dissipation of the spatially homogeneous Landau equation with hard potentials in terms of the entropy itself. We deduce from this explicit estimates on the speed of convergence towards equilibrium for the solution of this equation. In the case of so-called overmaxwellian potentials, the convergence is exponential. We also compute a lower bound for the spectral gap of the associated linear operator in this setting.     

Surgery and the Spinorial τ-Invariant

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then τ(N) ≥ min{τ(M), Λ _n }, where Λ _n is a positive constant depending only on n = dim M. Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ _n . Also, below Λ _n the values of τ cannot accumulate from above when varied over all manifolds of dimension n.     

Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.     

Two-dimensional thermometry using temperature-induced line shifts of ZnO:Zn and ZnO:Ga fluorescence

Sensitive temperature-induced line shifts of the near-band-edge emission from ZnO:Ga and ZnO:Zn are investigated for two-dimensional (2D) thermometry with nanosecond time resolution. Spectral and temporal concerns for 2D measurements and the feasibility for utilizing the line shifts for temperature measurements using a spectral ratio are investigated. Owing to the high temperature sensitivity, a precision of 1% at 800 K is reported for spectral ratio measurements. The technique is demonstrated by 2D measurements of the liquid temperature of burning methanol droplets.