一类退化椭圆型方程边值问题的适定性

研究一类特殊退化椭圆型方程边值问题的适定性,该类问题与双曲空间中的极小图的Dirichlet问题,曲而的无穷小等距形变刚性问题等等的研究密切相关,而这类方程的特征形式在区域上是变号的,其适定性是值得深入讨论的.最后,得到这类边值问题的H1弱解的存在性和唯一性.     

一类Kuramoto Sivashinsky方程的显式解

本文研究了一类KS方程的初边值问题.利用一致变换方法,并结合Green公式和Jordan引理,在半直线上得到了这类方程的显式解公式.所得结论将为该类方程适定性和数值计算的研究提供新的思路.     

主算子具非平凡核的抽象边值问题的适定性

本研究主算子具非平凡核时抽象边值问题的适定性问题，在较为一般的条件下，我们证明了抽象边值问题是适定的。我们还研究了无穷远处边界条件的改变引起抽象边值问题的出现多解问题，并就两种不同的无穷远处边界条件讨论了方程的多解。     

无粘、可压、绝热流体的Euler方程初值问题的适定性

根据分层理论提供的基本方法,讨论Euler方程的初值问题的适定性,给出了方程的典型初边值问题适定性的判别条件,确定了Euler方程的局部(准确)解的解空间构造,对适定问题给出了解析解的计算公式．     

一类二阶退化半线性椭圆型方程边值问题的适定性及解的正则性

本文考虑一类二阶退化半线性椭圆型方程边值问题.由椭圆正则化方法建立能量不等式,利用紧性推理,Banach—Saks定理,弱解与强解一致性,解常微分方程,椭圆型方程正则性定理,迭代方法.极值原理和Fredholm—Riesz-Schauder理论,可得相应线性问题适定性及解的高阶正则性;再由Moser引理和Banach不动点定理可得半线性问题解的存在性.这类问题与几何中无穷小等距形变刚性问题密切相关,其高阶正则性解的存在性对几何应用尤为重要.     

拟线性双曲抛物偶合组混合问题的局部古典解的存在性和唯一性

§1.引言 工程物理中的许多问题在数学上可归结为双曲抛物偶合方程组,如幅射流体力学方程组、粘性可压缩流方程组等。由于实践的需要使许多学者重视这类方程的研究,李大潜等在中对拟线性双曲抛物偶合组的第二边值问题利用Leray-Schauder不动点原理证得了局部古典解的存在唯一性。另外,在微分方程的适定性证明中除了用泛     

一类二阶退化半线性椭圆型方程边值问题的适定性及解的正则性

本文考虑一类二阶退化半线性椭圆型方程边值问题.由椭圆正则化方法建立能量不等式,利用紧性推理,Banach-Saks定理,弱解与强解一致性,解常微分方程,椭圆型方程正则性定理,迭代方法,极值原理和Fredholm-Riesz-Schauder理论,可得相应线性问题适定性及解的高阶正则性;再由Moser引理和Banach不动点定理可得半线性问题解的存在性.这类问题与几何中无穷小等距形变刚性问题密切相关,其高阶正则性解的存在性对几何应用尤为重要.     

Laplace方程混合边值问题的一个注记

本文讨论了一个角形区域上边界条件有间断点的Laplace方程混合边值问题 的适定性,并给出了完整的解答.     

关于特征情形的部分亚椭圆定理及其应用

<正> 边值问题的适定性和其解的正则性有紧密的联系.在研究边值问题解到边界的正则性时,Hrmander([2])的部分亚椭圆定理起着十分重要的作用.在研究蜕缩椭圆、双曲和混合型算子的边值问题时,提出了特征情形下的部分亚椭圆定理的问题.考虑Ω=R~(n-1)X(0,1)上的一阶算子方程.     

一类四阶差分方程边值问题及分形曲面的生成

本文研究一个二变元四阶差分方程边值问题，证明了此问题的适定性，揭示了解的结构。经过证明和数值模拟，可以作为分形曲面生成和插值的一种新方法。     

Well-Posed and Regular Nonlocal Boundary-Value Problems for Partial Differential Equations

The present paper deals with the well-posedness and regularity of one class of one-dimensional time-dependent boundary-value problems with global boundary conditions on the entire time interval. We establish conditions for the well-posedness of boundary-value problems for partial differential equations in the class of bounded differentiable functions. A criterion for the regularity of the problem under consideration is also obtained.     

Scalarization for pointwise well-posed vectorial problems

The aim of this paper is to develop a method of study of Tykhonov well-posedness notions for vector valued problems using a class of scalar problems. Having a vectorial problem, the scalarization technique we use allows us to construct a class of scalar problems whose well-posedness properties are equivalent with the most known well-posedness properties of the original problem. Then a well-posedness property of a quasiconvex level-closed problem is derived.      

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

Well-posedness of initial value problems for singular parabolic equations

We study well-posedness of initial value problems for a class of singular quasilinear parabolic equations in one space dimension. Simple conditions for well-posedness in the space of bounded nonnegative solutions are given, which involve boundedness of solutions of some related linear stationary problems. By a suitable change of unknown, the above results can be applied to classical initial-boundary value problems for parabolic equations with singular coefficients, as the heat equation with inverse square potential.     

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$ -regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty $ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.     

Numerical simulation and linear well-posedness analysis for a class of three-phase boundary motion problems

We investigate the linear well-posedness for a class of three-phase boundary motion problems and perform some numerical simulations. In a typical model, three-phase boundaries evolve under certain evolution laws with specified normal velocities. The boundaries meet at a triple junction with appropriate conditions applied. A system of partial differential equations and algebraic equations (PDAE) is proposed to describe the problems. With reasonable assumptions, all problems are shown to be well-posed if all three boundaries evolve under the same evolution law. For problems involving two or more evolution laws, we show the well-posedness case by case for some examples. Numerical simulations are performed for some examples.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Boundary controllability for conservative PDEs

Boundary observability and controllability problems for evolution equations governed by PDEs have been greatly studied in the past years. However, the problems were studied on a case-by-case basis, only for some particular types of boundary controls, and, moreover, several unnatural restrictions concerning lower-order terms were used.Our goal here is to give a general approach for boundary controllability problems, which is valid for all evolution PDEs of hyperbolic or ultrahyperbolic type, all boundary controls for which the corresponding homogeneous problem is well-posed, and all well-posedness spaces for the homogeneous problem. The first example of such equations is the class of hyperbolic equations, but valid examples are also equations such as the Schroedinger equation and various models for the plate equation.This work is essentially based on some apriori estimates of Carleman's type obtained by the author in a previous paper [29].This research was partially supported by the National Science Foundation under Grant NSF-DMS-8903747.     

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.