Rough path properties for local time of symmetric α stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.  The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ enables us to define the integral of the local time ${\int }_{?\infty }^{\infty }{?}_{?}^{\alpha ?1}f\left(x\right)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-variation with $1\le q<\frac{2}{3?\alpha }$. When $q\ge \frac{2}{3?\alpha }$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3?\alpha }\le q<4$.     

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.     

A cohomological study of local rings of embedding codepth 3

The generating series of the Bass numbers ${\mu }_R^i={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $e?d\le 3$. For each such $R$ it is proved that there is a real number $\gamma >1$, such that ${\mu }_R^{d+i}\ge \gamma {\mu }_R^{d+i?1}$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where ${\mu }_R^{d+2}={\mu }_R^{d+1}=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.     

Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

In this paper, we study the following quasilinear Schrödinger equation $-\Delta u+u-\Delta \left(u^2\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$where $N\ge 3$, $2^1=\frac{2N}{N-2}$, $h$ is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^4}=\infty$, we only need to assume that ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^2}=\infty$.     

Energy dissipation in the Smagorinsky model of turbulence

The Smagorinsky model often severely over-dissipates flows and, consistently, previous estimates of its energy dissipation rate blow up as $Re\to \infty$. This report estimates time averaged model dissipation, $〈{\epsilon }_S〉$, under periodic boundary conditions as$〈{\epsilon }_S〉\le 2\frac{U^3}{L}+{\mathrm{Re}}^{-1}\frac{U^3}{L}+\frac{32}{27}{C}_S^2{\left(\frac{\delta }{L}\right)}^2\frac{U^3}{L}\text{,}$ where $U,L$ are global velocity and length scales and $C_S\simeq 0.1,\delta <1$ are model parameters. Thus, in the absence of boundary layers, the Smagorinsky model does not over dissipate.