Jose Ruidival dos Santos Filho, Joaquim Tavares
Let $(\mathrm{M}, ω_{0})$ be a connected paracompact smooth oriented manifold. We establish a necessary and sufficient conditions on Lie subalgebra $\mathfrak{a}$ of $\mathrm{T M}$ such that its orbits becomes diffeomorphic to an open convex set of $\mathbb{R}^{n}$, where $n$ is the orbit dimension.
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Globally solvable complexes of pseudo-differential operators on the torus
本文研究环面上与超定算子组相关的伪微分算子复形，在常系数情形下刻画其全局可解性，并建立该复形与其正规形式在全局可解性上的联系。
Globally solvable time-periodic evolution equations in Gelfand-Shilov classes
该研究针对系数依赖时间与空间的周期发展算子，利用自伴全局椭圆算子的特征函数展开刻画 Gelfand–Shilov 空间，并给出其全局可解的充要条件。
Let $G_1$ and $G_2$ be compact Lie groups, $X_1 \in \mathfrak{g}_1$, $X_2 \in \mathfrak{g}_2$ and consider the operator \begin{equation*} L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), \end{equation*} where $a$ and $q$ are ultradifferentiable functions in the sense of Komatsu, and $a$ is real-valued. We characterize completely the global hypoellipticity and the global solvability of $L_{aq}$ in the sense of Komatsu. For this, we present a conjugation between $L_{aq}$ and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypo…
On globally smooth oscillating solutions of non-strictly hyperbolic systems
本文构造了一类非严格双曲拟线性系统，其柯西问题的振荡解在零稳态附近全局光滑，且振荡周期与拉格朗日轨迹初始点无关，并探讨了其在冷等离子体物理中的可实现性。
It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given solution, will be globally smooth as well. The examples of the solutions are constructed.
The bipolar non-isentropic compressible Euler-Maxwell system is investigated in $R^3$ in the present paper, and the $L^q$ time decay rate for the global smooth solution is established. It is shown that the total densities, total temperatures and magnetic field of two carriers converge to the equilibrium states at the same rate $(1+t)^{-3/2+3q/2}$ in $L^q$ norm. But, both the difference of densities and the difference of temperatures of two carriers decay at the rate $(1+t)^{-2-\frac{1}{q}}$, and the velocity and electric field decay at the rate $(1+t)^{-3/2+\frac{1}{2q}}$. This phenomenon on t…
We prove the global existence of smooth solution to the relativistic string equation in a class of data that is not small. Our solution admits the feature that the right-travelling wave can be large and the left-travelling wave is sufficiently small, and vice versa. In particular, the large-size solution exists in the whole space, instead of a null strip arising from the short pulse data. This generalizes the result of Liuli-Yang-Yu (Adv. Math. 2018) to the quasilinear setting with non-small data. In addition, in our companion paper, we are able to show the global solution here can also be see…
It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing, that there exist unique solutions which gain one derivative, globally. On the other hand, if the forcing term smooth enough, it is known that the solution gains two derivatives globally. In this paper, we explore classical techniques to show that if the force is sufficiently smooth, then the solution gains two derivatives globally. These methods break down when the force becomes singular. In this scenario, we use a Littlewood-Paley decomposition in Fourier space to show that a solutions gain two deriva…
The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander's condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth …
Unique inversion of smooth nonlinearity in semilinear wave equations on Lorentzian manifolds
研究洛伦兹流形上半线性波动方程光滑非线性项的恢复，核心方法是结合高阶线性化与扭曲平面波、高斯光束，利用多波相互作用恢复泰勒系数。
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
该研究针对具有全局可积测地流的黎曼流形上的半线性哈密顿PDE，基于正规形方法证明了小光滑解的几乎全局存在性抽象结果，并将其应用于带卷积势的NLS方程、非线性梁方程，同时证明了NLS基态的$H^s$稳定性。
Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
本文证明两个互补的小数据稳定性定理，分别给出球对称3D可压缩欧拉方程激波形成解的MGHD唯一存在性及带$1/r$密度尾部数据的全时空光滑全局存在性。
We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -Δu + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ Ω\subset \mathbb{R}^{n}$, where $ F : \mathbb{C} \rightarrow \mathbb{C} $ is Lipschitz. The same method gives a solution to $ - Δ_{g} u + F(u) = f $ for these boundary conditions on a smooth, compact Riemannian manifold $ (M, g) $ with $ \mathcal{C}^{1} $ boundary, where $ - Δ_{g} $ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version …
In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function $\hat p$ that is $\log$--Hölder continuous and an electric field $\mathbf{E}$ for that $\vert\mathbf{E}\vert^2$ is bounded and smooth. Note that these conditions do not imply that the vari…
Removable sets for pseudoconvexity for weakly smooth boundaries
研究 C^{1,1} 弱光滑边界上拟凸性可去集的判定条件，证明零测闭集外局部拟凸可推出全局拟凸，并以峰集型假设给出 Grauert–Remmert 结果的新证明。
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
Globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking
本文研究了一类超越波浪破碎的双分量非线性色散波方程，通过引入新变量将其转化为等价半线性系统，并利用常微分方程理论和逆变换方法，证明了原系统全局保守弱解的存在性。
We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.
A Lorentzian splitting theorem for continuously differentiable metrics and weights
本文通过椭圆技术与线适应曲线，将洛伦兹分裂定理推广至$C^1$度量与权重的全局双曲加权时空。
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