Several analytic inequalities in some Q−Q-spaces

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$Comment: 13 pages submitte

Riesz transforms on Q-type spaces with application to quasi-geostrophic equation

In this paper, we prove the boundedness of Riesz transforms $\partial_{j}(-\Delta)^{-1/2}$ ($j=1,2,...,n$) on the Q-type spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_{\alpha}^{\beta,-1}(\mathbb{R}^{2}).$Comment: 18 pages, submitte

On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves

In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, the existence of an explicit infinite family of quadratic twists with analytic rank $0$. In addition, we establish the $2$-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch--Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over $\mathbb{Q}$ without complex multiplication.Comment: 21 pages, including examples of full BSD conjecture, to appear in the Journal of the London Mathematical Societ

Generalized Birkhoff theorem and its applications in mimetic gravity

There is undetermined potential function $V(\phi)$ in the action of mimetic gravity which should be resolved through physical means. In general relativity(GR), the static spherically symmetric(SSS) solution to the Einstein equation is a benchmark and its deformation also plays a crucial role in mimetic gravity. The equation of motion is provided with high nonlinearity, but we can reduce primal nonlinearity to a frequent Riccati form in the SSS case of mimetic gravity. In other words, we obtain an expression of solution to the functional differential equation of motion with any potential function. Remarkably, we proved rigorously that there is a zero point of first order for the metric function $\beta(r)$ if another metric function $\alpha(r)$ possesses a pole of first order within mimetic gravity. The zero point theorem may be regarded as the generalization of Birkhoff theorem $\alpha\beta=1$ in GR. As a corollary, we show that there is a modified black hole solution for any given $V(\phi)$, which can pass the test of solar system. As another corollary, the zero point theorem provides a dynamical mechanism for the maximum size of galaxies. Especially, there are two analytic solutions which provide good fits to the rotation curves of galaxies without the demand for particle dark matter.Comment: 11 page

Multi-Stage Robust Transmission Constrained Unit Commitment: A Decomposition Framework with Implicit Decision Rules

With the integration of large-scale renewable energy sources to power systems, many optimization methods have been applied to solve the stochastic/uncertain transmission-constrained unit commitment (TCUC) problem. Among all methods, two-stage and multi-stage robust optimization-based methods are the most widely adopted ones. In the two-stage methods, nonanticipativity of economic dispatch (ED) decisions are not considered. While in multi-stage methods, explicit decision rules (for example, affine decision rules) are usually adopted to guarantee nonanticipativity of ED decisions. With explicit decision rules, the computational burden can be heavy and the optimality of the solution is affected. In this paper, a multi-stage robust TCUC formulation with implicit decision rules is proposed, as well as a decomposition framework to solve it. The solutions are proved to be multi-stage robust and nonanticipativity of ED decisions is guaranteed. Meanwhile, a computationally efficient time-decoupled solution method for the feasibility check subproblems is also proposed such that the method is suitable for large-scale TCUC problems with uncertain loads/renewable injections. Numerical tests are conducted on the IEEE 118-bus system and Polish 2383-bus system. Performances of several state-of-the-art methods are compared

On the Euler-Poincar\'e equation with non-zero dispersion

We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae Liu}. Our analysis exhibits some new concentration mechanism and hidden monotonicity formula associated with the Euler-Poincar\'e flow. In particular we show the abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.Comment: 18 page

Global well-posedness for the full compressible Navier-Stokes equations

In this paper, we mainly study the Cauchy problem for the full compressible Navier-Stokes equations in Sobolev spaces. We establish the global well-posedness of the equations with small initial data by using Friedrich's method and compactness arguments.Comment: arXiv admin note: text overlap with arXiv:1407.4661, arXiv:1109.5328 by other author

Robust Transmission Constrained Unit Commitment:A Column Merging Method

With rapid integration of power sources with uncertainty, robustness must be carefully considered in the transmission constrained unit commitment (TCUC) problem. The overall computational complexity of the robust TCUC methods is closely related to the vertex number of the uncertainty set. The vertex number is further associated with 1) the period number in the scheduling horizon as well as 2) the number of nodes with uncertain injections. In this paper, a column merging method (CMM) is proposed to reduce the computation burden by merging the uncertain nodes, while still guar-anteeing the robustness of the solution. By the CMM, the transmission constraints are modified, with the parameters obtained based on an analytical solution of a uniform approximation problem, so that the computational time is negligi-ble. The CMM is applied under a greedy-algorithm based framework, where the number of merged nodes and the ap-proximation error can be well balanced. The CMM is designed as a preprocessing tool to improve the solution efficiency for robust TCUC problems and is compatible with many solution methods (like two-stage and multi-stage robust optimi-zation methods). Numerical tests show the method is effective

Features of Motion Around Charged D-Stars

The motion of light and a neutral test particle around the charged D-star has been studied. The difference of the deficit angle of light from the case in asymptotically flat spacetime is in a factor $(1-\epsilon^2)$. The motion of a test particle is affected by the deficit angle and the charge. Through the phase analysis, we prove the existence of the periodic solution to the equation of motion and the effect of the deficit angle and the charge to the critical point and its type. We also give the conditions under which the critical point is a stable center and an unstable saddle point.Comment: 9 pages, 2 figure

On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for $p\in(1,4)$ and $a\in\dot{B}_{p,1}^{\frac2p}(\mathbb{R}^2)$ that the solution mapping $\mathcal{H}_a:F\mapsto\nabla\Pi$ to the 2-D elliptic equation $\mathrm{div}\big((1+a)\nabla\Pi\big)=\mathrm{div} F$ is bounded on $\dot{B}_{p,1}^{\frac2p-1}(\mathbb{R}^2)$. More precisely, we prove that $$\|\nabla\Pi\|_{\dot{B}_{p,1}^{\frac2p-1}}\leq C\big(1+\|a\|_{\dot{B}_{p,1}^{\frac2p}}\big)^2\|F\|_{\dot{B}_{p,1}^{\frac2p-1}}.$$ The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15]-[17]. When the viscosity coefficient $\mu(\rho)$ is a positive constant, we prove that (1.2) is globally well-posed