Parabolic equations with singular divergence-free drift vector fields

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;\xi,\tau)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm $\left\Vert u\right\Vert _{L^{\infty}(\textrm{BMO}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $\left(L-\partial_{t}\right)v=0$ are established for initial data in $L^{2}$-space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form.Comment: 28 page

Markov semi-groups generated by elliptic operators with divergence-free drift

In this paper we construct a conservative Markov semi-group with generator $L=\Delta+b\cdot\nabla$ on $\mathbb{R}^n$, where $b$ is a divergence-free vector field which belongs to $L^{2}\cap L^{p}$ with $\frac{n}{2}<p$. The research is motivated by the question of understanding the blow-up solutions of the fluid dynamic equations, which attracts a lot of attention in recent years.Comment: 12 page

Parabolic equations with divergence-free drift in space LtlLxqL_{t}^{l}L_{x}^{q}

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $\gamma=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $l=\infty,q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}\left(0,T;L^{n}\left(\mathbb{R}^{n}\right)\right)$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Semenov, 2006]. One of the results obtained in the present paper establishes the Aronson type estimate for critical and supercritical cases and for vector fields $b$ which are divergence-free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition enters the study of parabolic equations rather recently, mainly due to the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_{t}$ appears naturally.Comment: 31 page

Association of interleukin 10 rs1800896 polymorphism with susceptibility to breast cancer: a meta-analysis.

Objective: To evaluate the correlation between interleukin 10 (IL-10) -1082A/G polymorphism (rs1800896) and breast cancers by performing a meta-analysis. Methods: The Embase and Medline databases were searched through 1 September 2018 to identify qualified articles. Odds ratios (OR) and corresponding 95% confidence intervals (CIs) were applied to evaluate associations. Results: In total, 14 case-control studies, including 5320 cases and 5727 controls, were analyzed. We detected significant associations between the IL10 -1082 G/G genotype and risk of breast cancer (AA + AG vs. GG: OR = 0.88, 95% CI = 0.80-0.97). Subgroup analyses confirmed a significant association in Caucasian populations (OR = 0.89, 95% CI = 0.80-0.99), in population-based case-control studies (OR = 0.87, 95% CI = 0.78-0.96), and in studies with ≥500 subjects (OR = 0.88, 95% CI = 0.79-0.99) under the recessive model (AA + AG vs. GG). No associations were found in Asian populations. Conclusions: The IL10 -1082A/G polymorphism is associated with an increased risk of breast cancer. The association between IL10 -1082 G/G genotype and increased risk of breast cancer is more significant in Caucasians, in population-based studies, and in larger studies

Analysis on Heavy Quarkonia Transitions with Pion Emission in Terms of the QCD Multipole Expansion and Determination of Mass Spectra of Hybrids

One of the most important tasks in high energy physics is search for the exotic states, such as glueball, hybrid and multi-quark states. The transitions $\psi(ns)\to \psi(ms)+\pi\pi$ and $\Upsilon(ns)\to \Upsilon(ms)+\pi\pi$ attract great attentions because they may reveal characteristics of hybrids. In this work, we analyze those transition modes in terms of the theoretical framework established by Yan and Kuang. It is interesting to notice that the intermediate states between the two gluon-emissions are hybrids, therefore by fitting the data, we are able to determine the mass spectra of hybrids. The ground hybrid states are predicted as 4.23 GeV (for charmonium) and 10.79 GeV (for bottonium) which do not correspond to any states measured in recent experiments, thus it may imply that very possibly, hybrids mix with regular quarkonia to constitute physical states. Comprehensive comparisons of the potentials for hybrids whose parameters are obtained in this scenario with the lattice results are presented.Comment: 16 pages, 2 figur