Huiayu Jian
De Giorgi conjectured in 1979 that if a sequence of parabolic functionals Gamma converges to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. This paper studies the Gamma convergence for a kind of parabolic functionals, and it supports the De Giorgi's conjecture.
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We construct examples of blowup from smooth data for complex-valued solutions to linear uniformly parabolic equations in dimension $n \geq 2$, which are exactly as irregular as parabolic energy estimates allow.
In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.
Delayed diffusion with measure-valued kernels in nonlinear parabolic equations
研究由有限符号测度核控制的带延迟扩散项的非线性抛物方程，通过有效当前算子结构假设和总记忆算子路径强制条件，证明弱解存在唯一性及稳定性，涵盖坍缩延迟原子，还验证了p - 拉普拉斯型例子假设。
We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the nonexistence/existence of global-in-time solutions, and show the discontinuity of this critical exponent at the point which the inhomogeneous term becomes independent of time variable.
In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $(p,δ)$-structure, for all $p>1$ and for all $δ>0$. In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting…
In this paper we establish weighted $L^{q}$-$L^{p}$-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin and Besov type, whose trace theory is a…
Insensitizing controls for a quasi-linear parabolic equation with diffusion depending on gradient of the state
本文针对扩散项依赖状态梯度的拟线性抛物型方程，利用对偶论证、Carleman估计和右逆映射定理证明了不敏感化控制的存在性，解决了Xu Liu在2012年留下的开放问题，并讨论了相关推广。
Sharp spectral-scale stability for parabolic equations with measure-valued delay
本文研究抛物型方程解算子对延迟测度的依赖性，通过加权谱和与实现恒等式，在Dirichlet拉普拉斯情形下得到尖锐的最坏情形稳定性模，并给出谱准则与半线性推广。
Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
本文为 Gelfand 三元组上的线性抛物型发展方程建立 Banach 值侧向 Stieltjes 微积分，处理后验状态和原子跳跃，证明乘积公式、重构定理和能量恒等式，并通过 Galerkin 与 Rothe 格式构造存在性，以带 Stieltjes 时间的热方程为例。
We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity with respect to u and prove Hölder continuity of the solutions outside of singular sets.
Convergence of a finite volume scheme and dissipative measure-valued-strong stability for a hyperbolic-parabolic cross-diffusion system
本文构建双曲-抛物交叉扩散系统的耗散测度值解理论框架，提出全离散有限体积格式并证明其收敛性与弱-强稳定性，结果基于Young测度理论与熵估计。
Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients
研究高阶抛物方程的De Giorgi-Moser-Nash估计，获得基本解的高斯上界和Hölder正则性估计。
Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$
本文推广Nittka不变性准则至Hilbert空间值$L_2$空间，应用于纯二阶椭圆系统，证明其半群在区间$[p_-,p_+]$内$L_p$-压缩。
$G$-convergence of elliptic and parabolic operators depending on vector fields
针对依赖向量场的散度型椭圆与抛物算子，本文利用补偿紧性理论，在缺乏仿射函数保证的情形下建立了 G-收敛（H-收敛）紧性结果。
We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.
We develop a maximal regularity approach in temporally weighted $L_p$-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal wei…
We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behavi…
We consider an abstract parabolic problem in a framework of maximal monotone graphs, possibly multi-valued with growth conditions formulated with help of an $x-$dependent $N-$function. The main novelty of the paper consists in the lack of any growth restrictions on the ${N}$--function combined with its anisotropic character, namely we allow the dependence on all the directions of the gradient, not only on its absolute value. This leads us to use the notion of modular convergence and studying in detail the question of density of compactly supported smooth functions with respect to the modular c…
Regularity results for elliptic and parabolic systems of partial differential equations
本文研究向量值函数上一阶耦合椭圆算子的柯西问题，通过证明半群空间导数的逐点估计，得到了 Hölder、Zygmund、Sobolev 和 Besov 空间中的正则性结果。
Convergence to equilibrium for linear parabolic systems coupled by matrix-valued potentials
本文研究由矩阵值势耦合的线性抛物型方程组在无正性假设下的收敛到平衡态问题，针对所有 $\ell^p$-耗散势，利用与 $L^p$-空间几何结构相关的谱理论方法解决了 $p \neq 2$ 时的困难情形。
We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal pseudodifferential operator generates an analytic semigroup both on Banach space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no condition on the Banach space is imposed. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-auton…
Asymptotic Behaviors of Global Solutions to Fourth-order Parabolic and Hyperbolic Equations with Dirichlet Boundary Conditions
本文研究了四阶抛物型和双曲型方程在狄利克雷边界条件下的全局解渐进行为，建立了解趋于平衡态的收敛性及收敛速率估计，并通过数值模拟验证结果。
The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains.
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that these mild solutions exist uniquely in suitably chosen functional classes.
We prove the existence of a solution for the second order system of partial differential equations $\partial_t f = ν\cdotΔf + g\cdot\nabla f + h\cdot f + k$ by a Montel space version of Arzelà--Ascoli and bound all Schwartz semi-norms. We find that for the Euler and the Navier--Stokes equations the vorticity remains a Schwartz function as long as the classical solution exists. Our approach is not affected by viscosity. It treats the hyperbolic Euler and the parabolic Navier--Stokes equation simultaneously.
We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{ö}lder continuous Lp valued functions for some p > 2.
We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems.
The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied.
Motivated by diffusion processes on metric graphs and ramified spaces, we consider an abstract setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside well-posedness, we discuss positivity, $L^\infty$-contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoy these properties if and only if so does its counterpart with time-independent boundary conditions. Furthermore, we prove continuous dependence of the solution to the parabolic problem on the boundary conditions in the…
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied
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 $L=νΔ+u(x,t)\cdot\nabla$, 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 $\varGamma(x,t;ξ,τ)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension…
This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions $d\geq 3$. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE.
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occu…
We give a new example of a measure-valued process without a density, which arises from a stochastic partial differential equation with a multiplicative noise term. This process has some unusual properties. We work with the heat equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number, and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2). The exponent 2 is critical in the following sense. For exponents less than 2, the equation has function-valued solutions, and for exponent…
We study a nonstandard formulation of the Neumann initial value problem \begin{equation} \begin{array}{rl} u_t(x,t) = Δϕ(u(x,t)), & x \in Ω\subseteq \mathbb{R}^k, \ t \in \mathbb{R} \label{abstract}\\ u(x,0) = u_0(x), & x \in Ω. \end{array} \end{equation} with Neumann boundary conditions. The function $ϕ\in C^1(\mathbb{R})$ is assumed to be decreasing either in a bounded interval $(u^-,u^+)$, or in an unbounded interval $(u^-,+\infty)$: under this hypothesis, the aforementioned problem is ill-posed and only allows for measure-valued solutions. Moreover, such solutions are in general not unique…
We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the corresponding SPDE and we study the asymptotic behavior of its solutions, depending on the small parameter. We show that a large deviations principle holds and we give an explicit description of the action functional.
We consider operator-valued boundary value problems in $(0,2π)^n$ with periodic or, more generally, $ν$-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains $(0,2π)^n\times V$ with $ν$-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal $L^q$-regularity for such problems.
Convergence Analysis of a Finite-Volume Scheme for a Microglia--Amyloid Chemotaxis Model with Measure-Valued Vascular Boundary Sources
针对阿尔茨海默病相关的小胶质细胞-淀粉样蛋白趋化模型，引入适配测度边界源的弱解概念，构造有限体积格式并证明其离散解收敛到连续问题的弱解。
Control-constrained parabolic optimal control problems on evolving surfaces - theory and variational discretization
针对演化曲面上的控制约束线性二次最优控制问题，证明了状态方程弱解的存在唯一性，并提出了变分离散化方法，证明了其收敛性，在弱正则性假设下获得了最优收敛阶。
In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetr…
Existence of parabolic minimizers to the total variation flow on metric measure spaces
本文在满足倍增条件与 Poincaré 不等式的度量测度空间上，利用抽象的 BV 值抛物函数空间和纯变分论证，证明全变差流 Cauchy–Dirichlet 问题抛物极小元的存在性。
In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand parabolic category $\mathcal{O}^\mathfrak{p}$ geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold $G/P$ for $…
Approximation of Random Evolution Equations of Parabolic type
本文提出了一种框架，用于研究随机抛物型方程的近似收敛性，通过多项式混沌展开和正则性条件，实现随机性和确定性下的高阶收敛。
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