求解一类Helmholtz方程Cauchy问题的中心差分正则化方法

Helmholtz方程Cauchy问题是严重不适定问题,本文我们在一个带形区域上考虑了一类Helmholtz方程Cauchy问题:已知Cauchy数据u(0,y)=g(y),在区间0＜x＜1上求解.我们用半离散的中心差分方法得到了这一问题的正则化解,给出了正则化参数的选取规则,得到了误差估计.     

Dynamics of droplets in an agitated dispersion with multiple breakage. Part II: uniqueness and global existence

In Part I (see Math. Methods Appl. Sci.) a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented. Our aim was to extend a previous version (see Proceedings of the Symposium Organized by the Sonderforschungsbereich 438 on the Occasion of Karl‐Heinz Hoffman's 60th Birthday, Lectures in Applied Mathematics, Springer: Berlin, 2000; 123–141) in order to describe the influence of each breakage mode. Here we complete the mathematical analysis to ensure the well posedness (in the sense of Hadamard) of the Cauchy problem for the main evolution equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Local well‐posedness and ill‐posedness for the fractal Burgers equation in homogeneous Sobolev spaces

In this paper, we consider local well‐posedness and ill‐posedness questions for the fractal Burgers equation. First, we obtain the well‐posedness result in the critical Sobolev space. We also present an unconditional uniqueness result. Second, we show the ill‐posedness from the point of view of the Picard iterations in the supercritical space. Copyright © 2008 John Wiley & Sons, Ltd.     

An operator approach for an oblique derivative boundary‐transmission problem

We consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.     

Unified Approaches to Well-Posedness with Some Applications

We present unified approaches to Hadamard and Tykhonov well-posedness. As applications, we deduce Tykhonov well- posedness for optimization problems, Nash equilibrium point problems and fixed point problems etc. Especially, by applying such approaches, we deal with the well- posedness as stated in (Lignola and Morgan (2000), Journal of Global Optimization 16, 57–67) in which Lignola and Morgan investigated directly and intensively Tykhonov types of well- posedness for optimization problems with constraints defined by variational inequalities, namely, generalized well- posedness and strong well- posedness. We give some sufficient conditions for Hadamard well- posedness of such problems and deduce relations between Hadamard type and Tykhonov type of well- posedness. Finally, as corollaries, we derive generalized well- posedness and strong well- posedness for these problems.     

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A note on the instability of a focusing nonlinear damped wave equation

We investigate the initial value problem for a nonlinear damped wave equation in two space dimensions. We prove local well‐posedness and instability by blow‐up of the standing waves. Copyright © 2014 John Wiley & Sons, Ltd.     

An optimal filtering method for the Cauchy problem of the Helmholtz equation

In this paper, we consider a Cauchy problem for the Helmholtz equation in a rectangle. An optimal filtering method is presented for approximating the solution of this problem, and the Hölder type error estimate is obtained. Numerical illustration shows that the method works effectively.     

On the well‐posedness for Keller–Segel system with fractional diffusion

In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier‐Herz spaces with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

Lipschitzian semigroups and abstract functional differential equations

The paper is devoted to studying an abstract functional differential equation by a nonlinear semigroup approach. We first prove in details the equivalence of the well posedness of an abstract functional differential equation and an associated abstract Cauchy problem in the sense of strong solutions. Secondly, a sufficient condition is derived for well posedness of the abstract functional differential equation. Thirdly, we present principles of linearized stability for the abstract functional differential equation. Finally, the results obtained are applied to a reaction-diffusion equation with delays.     

The stability of localized solutions of Landau‐Lifshitz equations

We study the Landau‐Lifshitz equation of ferromagnetism on ?², with an easy‐axis anisotropy. We establish the existence of topologically nontrivial, periodic solutions, and show they are stable against equivariant perturbations. Along the way, we establish the global well‐posedness of the Cauchy problem for a class of data with no size restriction. © 2002 Wiley Periodicals, Inc.     

A linear sampling method for inverse problems of diffraction gratings of mixed type

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.     

The Carleman formula for the Helmholtz equation on the plane

We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.     

Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally smooth. Afterward, we deal with discontinuous entropy solutions, and obtain the global well posedness of the Cauchy problem in this class. Eventually, we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data. As a byproduct, we establish the existence of an utility process which is a solution to a backward-forward stochastic differential equation, for a given class of final utilities, which is relevant for financial applications.     

A backward problem for the time‐fractional diffusion equation

In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time‐fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill‐posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem

This Note is concerned with the severely ill-posed Cauchy–Helmholtz problem. This Cauchy problem being rephrased through an “interfacial” equation, we resort to an Aitken–Schwarz method for solving this equation. Numerical trials highlight the efficiency of the present method.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

Local well‐posedness for the Cauchy problem of the MHD equations with mass diffusion

This paper studies the Cauchy problem of the MHD equations with mass diffusion. We use the Tikhonov fixed point theorem to prove a local‐in‐time well‐posedness theorem. Copyright © 2010 John Wiley & Sons, Ltd.     

Well‐posedness for the Cauchy problem of the modified Hunter–Saxton equation in the Besov spaces

This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces H^s) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.