An optimal gap theorem

By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for K\"ahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius $r$ centered at any fixed point $o$ is a function of $o(r^{-2})$. Furthermore via a relative monotonicity estimate we obtain a stronger statement, namely a `positive mass' type result, asserting that if $(M, g)$ is not flat, then $\liminf_{r\to \infty} \frac{r^2}{V_o(r)}\int_{B_o(r)}\mathcal{S}(y)\, d\mu(y)>0$ for any $o\in M$

Global Stability of a Premixed Reaction Zone (Time-Dependent Liñan’s Problem)

Global stability properties of a premixed, three-dimensional reaction zone are considered. In the nonadiabatic case (i.e., when there is a heat exchange between the reaction zone and the burned gases) there is a unique, spatially one-dimensional steady state that is shown to be unstable (respectively, asymptotically stable) if the reaction zone is cooled (respectively, heated) by the burned mixture. In the adiabatic case, there is a unique (up to spatial translations) steady state that is shown to be stable. In addition, the large-time asymptotic behavior of the solution is analyzed to obtain sufficient conditions on the initial data for stabilization. Previous partial numerical results on linear stability of one-dimensional reaction zones are thereby confirmed and extended

Well-posedness for a model of individual clustering

25 pagesInternational audienceWe study the well-posedness of a model of individual clustering. Given p > N ≥ 1 and an initial condition in W 1,p (Ω), the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if N = 1, as well as, the convergence to steady states for one of these rates

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters

Scalar Field Quantization Without Divergences In All Spacetime Dimensions

Covariant, self-interacting scalar quantum field theories admit solutions for low enough spacetime dimensions, but when additional divergences appear in higher dimensions, the traditional approach leads to results, such as triviality, that are less than satisfactory. Guided by idealized but soluble {\it non}renormalizable models, a nontraditional proposal for the quantization of covariant scalar field theories is advanced, which achieves a term-by-term, divergence-free, perturbation analysis of interacting models expanded about a suitable pseudofree theory, which differs from a free theory by an O(\hbar^2) counterterm. These positive features are realized within a functional integral formulation by a local, nonclassical, counterterm that effectively transforms parameter changes in the action from generating mutually singular measures, which are the basis for divergences, to equivalent measures, thereby removing all divergences. The use of an alternative model about which to perturb is already supported by properties of the classical theory, and is allowed by the inherent ambiguity in the quantization process itself. This procedure not only provides acceptable solutions for models for which no acceptable, faithful solution currently exists, e.g., \phi^4_n, for spacetime dimensions n\ge4, but offers a new, divergence-free solution, for less-singular models as well, e.g., \phi^4_n, for n=2,3. Our analysis implies similar properties for multicomponent scalar models, such as those associated with the Higgs model.Comment: 45 pages, has relevance for the Higgs model, review and updated analysis, version accepted for publicatio