Connor Mooney
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.
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Asymptotic solutions of a quasilinear parabolic equation with a small parameter at the higher derivative are constructed near large-gradient and Lagrange singularities of A-type, which represent interest for studying processes of shock waves formation in physical media with a small nonzero viscosity.
The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds. Such a theory is of importance for the study of elliptic and parabolic equations on non-compact, or even incomplete manifolds, with or without boundary.
Gamma-convergence of integral functionals depending on vector-valued functions over parabolic domains
De Giorgi在1979年指出，如果一个抛物泛函序列Gamma收敛到一个极限泛函，那么相应的梯度流在适当改变时间尺度后也会收敛。本文研究了一类抛物泛函的Gamma收敛性，证明了De Giorgi猜想.
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a-priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is als…
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the conservation law $u_{t}+[H(u)]_x=0$. After having proved a precise statement of the formal relation $U_x=u$, we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.
Regularity and existence for a mixed local-nonlocal parabolic equation with variable singularities and measure data
该研究针对含奇异非线性项与测度数据的混合局部-非局部抛物问题，引入可变奇异指数，证明了弱解的存在性、非存在性、正则性及渐近行为，相关结果对纯局部算子也成立。
Relaxation of the flow of triods by Curve Shortening Flow via the vector-valued parabolic Allen-Cahn equation
本文给出了一类向量值抛物型Allen-Cahn方程的解$u_ε$，该方程以$ε\to 0$为界面，在曲线缩短流下演化为给定的三角形.
This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the Hölder-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a Kähler manifold, this Hölder estimate works when the metrics possess conic singularities along a normal crossing divisor.
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。
Zeros of complex caloric functions and singularities of complex viscous Burgers equation
我们表明，考虑复值函数的一维粘性Burgers方程的发展有限时间奇点从companies支持的光滑数据。通过Cole-Hopf变换，将解的奇性与热传导方程复值解的零点联系起来。我们证明，这样的零是孤立的，如果他们不存在于初始数据。
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…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.
Existence and uniqueness for renormalized solutions to a general noncoercive nonlinear parabolic equation
本文引入包含奇性与无界低阶项的一般非强制非线性抛物问题的重整化解概念，并证明其存在性与唯一性。
We study (higher order) asymptotic behaviour near branch points of stationary $n$-dimensional two-valued $C^{1, μ}$ graphs in an open subset of ${\mathbb R}^{n+m}$. Specifically, if $M$ is the graph of a two-valued $C^{1, μ}$ function $u$ on an open subset $Ω\subset {\mathbb R}^{n}$ taking values in the space of un-ordered pairs of points in ${\mathbb R}^{m}$, and if the integral varifold $V = (M, θ),$ where the multiplicity function $θ\, : \, M \rightarrow \{1, 2\}$ is such that $θ=2$ on the set where the two values of $u$ agree and $θ=1$ otherwise, is stationary in $Ω\times {\mathbb R}^{m}$ …
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。
Global Hölder Solvability of parabolic equations on domains with capacity density conditions
本文研究了具有容量密度条件的领域上抛物型方程的全局Hölder解的存在性，允许数据在边界附近具有临界奇异性。
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。
Formation and Behavior of Dirac Singularities in the Parabolic-Elliptic Keller-Segel System in Dimensions $n\geq 3$
研究抛物-椭圆凯勒-塞格尔系统中非负径向对称解的形成与行为，揭示狄拉克奇点的形成机制及渐进行为。
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。
In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapp…
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。
Regularization for fractional integral. Application to nonlinear equations with singularities
本文给出了分数次积分的δ序列正则化，并应用它得到了Colombeau代数中具有奇异性的非线性方程组的存在唯一性定理：具有极核的非线性积分方程组和具有强奇异初值和非Lipschitz非线性项的非线性抛物型方程（具有非线性保守项和Schrödinger核的普通型）.在非线性抛物型方程的情形下，我们实际上对热半群进行了关于时间变量$t.$的δ序列正则化。我们对线性薛定谔方程也进行了同样的正则化。
Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions are in fact more regular, and examples of systems with weak solutions which develop singularities in finite time. Our main result is the extension of a regularity result due to Kalita to the stochastic case. Concerning the examples with singular solutions (outside the setting of Kalita's regularity result), we do not know whether stochastic noise may prevent …
Singular and regular solutions of a non-linear parabolic system
我们研究了一个耗散非线性方程的Navier-斯托克斯方程的某些功能。我们证明了径向对称紧支撑的初始数据的演化不会导致维数$n\leq 4$的奇异性。对于尺寸$n>4$，我们提出了强有力的数值证据支持爆破解决方案的存在。此外，使用相同的技术，我们数值证实了Lepin关于存在自相似奇异解的半线性热方程的一个猜想。
In this paper we study the equation $Lu=f$, where $L$ is a $\C$-valued vector field in $\R^2$ with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers attached to the vector field.
Global comparison principles for the $p$-Laplace operator on Riemannian manifolds
我们证明了$p$ -抛物流形上的$p$ -Laplacian的整体比较结果。这些涉及实值和向量值映射与有限的$p$ -能量。
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…
In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second author shows that two-valued $C^{1, μ}$ harmonic functions on $n$ dimensional domains, which are typically-non-minimizing stationary points of Dirichlet energy, play an essential role in the study of multiplicity 2 branch points of stable codimension 1 rectifiable currents of dim…
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.
Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting
本文建立余维数为一时环面值映射的 $W^{1,1}$ 能量、BV 提升能量与 normal/integral 流 Plateau 最小质量之间的等价关系，并给出积分 Plateau 问题解的界。
Construction of eternal solutions for a semilinear parabolic equation
抛物型方程的永恒解（那些定义为所有时间的方程）通常是相当罕见的。例如，热方程只有一个永恒的解-平凡解。虽然热方程的解存在于所有向前的时间，但它们不能在时间上向后延伸。非线性会使情况变得更糟，因为解在向后和向前的时间中都可能形成奇点。然而，半线性抛物型方程也可以支持非平凡的永恒解。本文展示了如何构造一个至少有两个不同平衡解的半线性方程的非平凡永恒解。由此产生的永恒的解决方案是一个异宿轨道连接两个给定的平衡。
Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we pre- dict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two …
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one n-dimensional half-planes that meet along an (n-1) - dimensional axis. The key ingredient is a new Excess Improvement Lemma obtai…
Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption
研究一类具有退化强吸收的非线性抛物型方程解的极限性态。我们证明了两种类型的现象可能发生：逐点奇点或形成剃刀刀片（或持久奇点）。
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…
In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multi-valued Dirichle…
In the theory of non-linear parabolic and elliptic partial differential equations, the notion of maximal regularity plays an essential role in establishing existence, regularity and boundedness of solutions. There is a long history of works where sufficient conditions for maximal regularity have been established: First scalar equations and systems of finitely many coupled equations have been considered. Around 2000, the vector-valued case with infinite-dimensional range space $E$ became accessible to the development and progress in theory of $\mathcal{R}$-bounded operator families and its clos…
In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where $B(x)$ is a uniformly ell…
In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.
On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces
本文研究了在奇混合范数Morrey- Sobolev空间中，具有奇异系数的抛物方程的可解性问题，通过引入混合范数Morrey类和BMO条件，证明了存在性和唯一性定理。
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。
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。
We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive $\mathcal{R}$-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.
Blow up analysis for a parabolic MEMS problem, I: Hölder estimate
本文针对抛物型 MEMS 方程的淬灭现象，运用爆破方法与 Liouville 定理建立解的最优 Hölder 估计及一致连续解序列的收敛理论，并据此证明破裂集分层定理。
We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation ıs obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
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.
In this paper we investigate the propagation of singularities in a nonlinear parabolic equation with strong absorption when the absorption potential is strongly degenerate following some curve in the $(x,t)$ space. As a very simplified model, we assume that the heat conduction is constant but the absorption of the media depends stronly of the characteristic of the media. More precisely we suppose that the temperature $u$ is governed by the following equation \label{I-1} \partial_{t}u-Δu+h(x,t)u^p=0\quad \text{in}Q_{T}:=R^N\times (0,T) where $p>1$ and $h\in C(\bar Q_{T})$. We suppose that $h(x,…
Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.
We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^α$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $α\in (-1, \infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direc…
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.
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…
We consider an inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE. In 2006, Cao and Pereverzev, used a \textit{natural linearisation} method for identifying a scalar valued diffusion coefficient in a parabolic PDE. In this paper, we make use of that idea for identifying a matrix valued coefficient, namely, using the notion of a weak solution for a parabolic PDE, we transform our non-linear inverse problem into a problem of solving an ill-posed operator equation where the operator depending on the data is linear. For the purpose of obtaining stable approx…
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