Oleg Zubelevich
We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.
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In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.
Fractional powers approach of maximal accretive operators for a class of third order abstract Cauchy problems
该研究将极大增生算子分数幂理论应用于时间三阶线性抽象演化方程，推导得到方程解的存在性、正则性及渐近行为相关结果。
In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $n\geqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.
In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of one-sided coupled operator matrices provides an excellent framework to study the well-posedness of such problems. We show that with this machinery even operator splitting methods can be treated conveniently and rather efficiently. We consider three specific examples: the Lie (sequential), the Strang and the weighted splitting, and prove the convergence of these …
Strip-type operators and abstract Cauchy problems
本文通过带状和抛物线型算子，在Banach空间中研究非齐次抽象线性Schrödinger和波动方程的经典解适定性，并推广到R-有界情形，最后应用于抽象半线性波动方程短时解的存在唯一性。
Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators
本文研究了具有分数幂的近似扇形算子的直接和反向抽象柯西问题，推导了显式解算子并探讨了时间变量系数的求解方法。
In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.
In this article we first derive some sufficient conditions to establish the monotonicity and comparison principles of the semi-flow generated by non-densely defined Cauchy problems. We apply our results to a class of age structured population models. As a consequence we obtain a monotone semi-flow theory and some comparison principles for age structured models.
In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. By applying this result, the Cauchy problem for systems of Boussinesq equations of finite or infinite orders are studied
Due to the seminal works of Hochbruck and Ostermann exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than $π/2$ (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing on the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the sem…
Let $M$ be a compact abstract $CR$ manifold of arbitrary $CR$ codimension. Under certain conditions on the Levi form we prove the infinite dimensionality of some global cohomology groups of $M$.
An adjoint control method for initial condition identification of the Abstract Cauchy problem
针对抽象柯西问题的初始条件识别，提出并分析了一种基于伴随方程控制的通用方法，通过分析和数值实验验证了其有效性。
In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we…
A general class of singular abstract Cauchy problems is considered which naturally arises in applications to certain Free Boundary Problems. Existence of an associated evolution operator characterizing its solutions is established and is subsequently used to derive optimal regularity results. The latter are well known to be important basic tools needed to deal with corresponding nonlinear Cauchy Problems such as those associated to Free Boundary Problems.
We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.
Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups
本研究给出局部凸向量格上非线性单调半群生成广义粘性解的$K$-凸性等条件，证明相关半群可生成抽象柯西问题的粘性解，并应用于两类无穷维过程的控制问题。
We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we can apply the Laplace transform to any $n+1$ dimensional linear higher-order partial differential equations $P(\partial_x,\partial_t)u=f(x,t)$ directly, without using the Fourier transform. By making introduction of abstract operators $G(\partial_x,t):=\mathcal{L}^{-1}[1/P(\partial_x,s)]$, the analytic solutions of initial value problems are expressed in th…
We study maximal regularity in interpolation spaces for the sum of three closed linear operators on a Banach space, and we apply the abstract results to obtain Besov and Hölder maximal regularity for complete second order Cauchy problems under natural parabolicity assumptions. We discuss applications to partial differential equations.
Maximal regularity estimates for the abstract Cauchy problems
本研究拓展Da Prato-Grisvard极大正则性理论，确定Banach空间中扇形算子相关插值空间，建立齐次与非齐次$L^1$-、加权$L^1$-极大正则性新结果，重构Kalton-Portal刻画条件，并提出非经典插值函子下$L^p$-极大正则性的新插值框架。
For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.
The abstract first order Cauchy problem is solved in terms of Taylor's series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order lower than the Cauchy problem are built-in to the problem, it suffices to solve the higher order derivatives in terms of the lower order ones, in a simple way.
The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
In this paper, the existence, the uniqueness and estimates of solution to the integral Cauchy problem for linear and nonlinear abstract wave equations are proved. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.
In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order $α\in(0,1]$, and discuss the continuity of the respective solutions regarding the parameter $α$. I also present a study about the continuity of the Mittag-Leffler families of operators (for $α\in(0,1]$), induced by sectorial operators.
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
Fiedler and Mallet-Paret prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are obtained. These problems have a numeros applications which occur in fluid mechanics, environmental engineering and atmospheric dispersion of pollutants.
Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation
本文针对由无界线性算子和 Caputo 分数阶导数定义的非齐次柯西问题，提出仅依赖齐次问题传播子的新温和解表示，统一了次抛物、抛物和次双曲情形，并在显著更弱的正则性假设下证明强收敛，适用于任意阶数 α∈(0,2) 的数值计算。
Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=Δu -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data.
We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by Hörmander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by Métivier). As an example of application, we apply our …
The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says that by assuming the regularity of initial data, asymptotic profiles (of arbitrary order) are explicitly written by using the semigroup $e^{-tA}$ generated by $-A$. To prove this, a kind of maximal regularity for $e^{-tA}$ is used.
In this paper, we prove local existence and uniqueness of analytic sharp-front solutions to a generalised SQG equation by the use of an abstract Cauchy--Kowalevskaya theorem. Here, the velocity is determined by $u = |\nabla|^{-2β}\nabla^\perpθ$ which (for $1<β\leq 2$) is more singular than in SQG. This is achieved despite the appearance of pseudodifferential operators of order higher than one in our equation, by recasting our equation in a suitable integral form. We also provide a full proof of the abstract version of the Cauchy--Kowalevskaya theorem we use.
In this paper, regularity properties, Strichartz type estimates to solutions of multipoınt Cauchy problem for linear and nonlinear abstract wave equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.
Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type
提出一种指数收敛的数值方法，基于新解公式和sinc求积逼近具有Caputo时间分数阶导数和无界算子系数的抽象Cauchy问题，适用于全范围阶数并支持多级并行，数值实验证实了理论误差估计。
In this paper, regularity properties of Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained.
In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representat…
General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in Gene…
Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces
本文研究横向Hilbert空间上带正自伴算子项的抛物方程在一致局部空间中的Cauchy问题，证明线性算子因定义域可能不稠密而非扇形算子，并借助抽象发展方程理论在弱版本空间中建立适定性。
In this paper, the regularity properties of Cauchy problem for linear and nonlinear nonlocal wave equations are studied.The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform are operator functions defined in Hilbert space H together with some growth conditions. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data and the operator functions. By selecting the space H and the operators, the wide class of wave equations in the field of physics are obtained.
For an elliptic complex of first order differential operators on a smooth manifold, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. T…
An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces
本文针对 Hilbert 空间中的抽象 Cauchy 问题提出了弱解的定义，并探讨了该弱解的存在性与唯一性结果。
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even tho…
We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the 3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available…
The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic …
We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order operator to show that it generates an analytic semigroup.
We consider a linear transport equation on the edges of a network with time-varying coefficients. Using methods for non-autonomous abstract Cauchy problems, we obtain well-posedness of the problem and describe the asymptotic profile of the solutions under certain natural conditions on the network. We further apply our theory to a model used for air traffic flow management.
An abstract framework for a class of nonlocal structured population models: existence, uniqueness and stability of steady states
本文提出了一类非局部结构人口模型的抽象框架，分析了稳态解的存在性、唯一性和稳定性，并通过反例探讨了稳定性条件的限制。
In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age and the space variables. We handle this linear problem by re-writing it as a non-densely defined abstract Cauchy problem. To that aim we develop a new result on the closedness of a commutative sum of two non-densely defined operators by using the theory of integrated semigroups. As an application of this abstract result, we are able to associate a suitable integrated semigroup to some age-structured problem with spatial diffusion and equipped with non-homogeneous bo…
The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann-Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional time-derivatives of order less than one are considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfy certain complete monotonicity properties. Based on these properties, the well-posedness of the problem is established and a series expansion of …
Covariant definition of Double Null Data and geometric uniqueness of the characteristic initial value problem
本文以完全微分同胚和规范协变的方式推广了双零数据的定义，证明其完备性，并引入抽象等距概念证明了特征柯西问题的几何唯一性。
Dynamic boundary conditions for time dependent fractional operators on extension domains
该研究针对非光滑域上带Wentzell型边界条件的含时分数阶算子抛物半线性非自治问题，通过演化族证明抽象柯西问题温和解的存在唯一性，再用广义分数阶格林公式验证其为原问题的解。
Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},…
We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem…
Non-autonomous semilinear fractional evolution equations: well-posedness and ultracontractivity results
针对满足Acquistapace-Terreni条件的时间依赖扇形算子的时间分数阶半线性抛物型抽象柯西问题，证明了温和解的局部存在性，并在适当初始条件下证明全局存在性，方法为证明相关分数阶演化族的超压缩性估计。
In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the …
Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface
本文研究具有界面的色散材料中非线性麦克斯韦方程的适定性与指数稳定性，通过演化方程框架在加权函数空间中建立适定性并给出实现指数稳定的条件。
Dirichlet problems associated to abstract nonlocal space-time differential operators
本文研究抽象分数阶时空算子 $(\partial_t + A)^s$ 的狄利克雷问题，提出 $L^p$-解概念并建立推广常数变易公式的温和解表示，同时与 Riemann-Liouville 和 Caputo 型初值问题的解概念进行比较。
On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model
本文研究了与Timoshenko模型相关的抽象非线性耦合双曲方程组的Cauchy问题，提出了一种对称的三层半离散时间步进方案，通过时间中点评估非线性项，将原问题转化为线性问题并实现并行计算，证明了方法的收敛性及时间步长的二阶精度。
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