常系数线性$q$-差分微分方程形式解的可和性与收敛性研究

A note on $G_q$-summability of formal solutions of some linear $q$-difference-differential equations

The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].

Exponential type Nagumo norms and summability of formal solutions of singular partial differential equations

In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in \CC^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

Almost Everywhere Strong Summability of Double Walsh-Fourier Series

In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation $(\frac{1}{n}\sum\limits_{m=0}^{n-1}\left\vert S_{mm}f - f \right\vert^{p})^{1/p}\rightarrow 0$ for every two-dimensional functions belonging to $L\log L$ and $0<p\le 2$. From the theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.

Analytic and summable solutions of inhomogeneous moment partial differential equations

We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient conditions for the convergence, analytic continuation and summability of formal power series solutions in terms of properties of the inhomogeneity. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z).

Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables

We investigate Gevrey order and 1-summability properties of the formal solution of a general heat equation in two variables. In particular, we give necessary and sufficient conditions for the 1-summability of the solution in a given direction. When restricted to the case of constants coefficients, these conditions coincide with those given by D.A. Lutz, M. Miyake, R. Schaefke in a 1999 article, and we thus provide a new proof of their result.

Existence, Uniqueness, Analyticity, and Borel Summability of Boussinesq and Magnetic Benard Equations

Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…

Schrödinger方程的小时间行为与可和性

Small Time Behavior and Summability for the Schrödinger Equation

本文研究了Schrödinger方程在小时间下的收敛行为与可和性，证明了在特定势条件下Sobolev空间中的最优结果，并探讨了s<1/4时局部化极大算子的L^p有界性问题。

Existence, Uniqueness, Analyticity, and Borel Summability for Boussinesq Equations

Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…

两个复时间变量问题中的参数形式 Gevrey 渐近展开

Parametric formal Gevrey asymptotic expansions in two complex time variable problems

本文研究复域中含两个复时间变量的奇异扰动偏微分方程的解析与形式解，利用 Borel 平面上的解析延拓克服可和性困难，并发现若干指数衰减率以建立解间的渐近层级。

Nonlinear evolution PDEs in R^+ \times C^d: existence and uniqueness of solutions, asymptotic and Borel summability

We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form $${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left( {\bf x}, t, \{\partial_{\bf x}^{\bf j} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) = {\bf {u}}_I({\bf x})$$ with $\mathbf{u}\in\CC^{r}$, for $ t\in (0,T)$ and large $|{\bf x}|$ in a poly-sector $S$ in $\mathbb{C}^d$ ($\partial_{\bf x}^{\bf j} \equiv \partial_{x_1}^{j_1} \partial_{x_2}^{j_2} ...\partial_{x_d}^{j_d}$ and $j_1+...+j_d\le n$). The principal part of the constant coefficient $n$-th order differential operator $\mathc…

源自熵近似的正则化运输成本

A regularized transportation cost stemming from entropic approximation

本文研究了在熵正则化下最优运输问题的收敛性，通过弱紧致性条件证明正则化函数的收敛，并探讨了多边际问题的应用。

Summable solutions of some partial differential equations and generalised integral means

We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some multidimensional partial differential equations in terms of holomorphic properties of generalised integral means of the Cauchy data.

Existence of a Stationary Navier-Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions

We consider the large time behavior of the Navier-Stokes flow past a rigid body in $\mathbb{R}^n$ with $n\geq 3$. We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow\infty$ in $L^q$ with $q\in[n,\infty]$. This is c…

二阶奇异摄动线性常微分方程精确 WKB 解的存在性与唯一性

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

本文研究复域内一般二阶奇异摄动线性 ODE 的精确 WKB 解，证明其在一般及闭合轨迹下的存在唯一性，并给出解基含义、Borel 可和性及 Borel 变换显式公式。

参数和随机椭圆偏微分方程的完全离散近似

Fully discrete approximation of parametric and stochastic elliptic PDEs

本文研究了参数和随机椭圆偏微分方程的完全离散近似，通过结合空间和参数近似性，利用Taylor、Jacobi和Hermite展开，获得收敛率与总自由度相关的近似结果。

Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions

In this paper, we study the mean field limit of interacting particles with memory that are governed by a system of interacting non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in an extended phase space. We obtain the fundamental solution (Green's function) for this equation, for the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction. Furthermore, for nonc…