Boundary problems for systems of partial differential operators of mixed order

On a compact n-dimensional C^∞ manifold $\text{Ω}$ with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

A Discrete Phenomenon in the C and Analytic Regularity of the Blow-Up Curve of Solutions to the Liouville Equation in One Space Dimension

We give a complete discussion of the C^∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the Lp-Lq estimates.     

On the nonlinear type singularities for semilinear Cauchy problemsSur les singularitiés de type non-linéaire pour des problèmes de Cauchy semi-linéaires

We consider the Cauchy problem for the semilinear wave equation. The Cauchy data are assumed to be conormal with respect to a point, and the source term is polynomial with respect to the solution and its first derivatives. Thanks to the study of multiplicative properties of some refined hyperbolic conormal spaces, we improve the known results about the nonlinear type singularities of the solution. To cite this article: D. Fang et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 453–458.     

Singular perturbation degenerate problems occurring in atmospheric dispersion of pollutants

The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued L_p-spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.     

A counter-example to the characterization of the discontinuous value function of control problems with reflectionUn contre-exemple à la caractérisation de la fonction-valeur discontinue d'un problème de contrôle réfléchi

We consider a finite horizon deterministic optimal control problem with reflection. The final cost is assumed to be merely a locally bounded function which leads to a discontinuous value function. We address the question of the characterization of the value function as the unique solution of an Hamilton–Jacobi equation with Neumann boundary conditions. We follow the discontinuous approach developed by Barles and Perthame for problems set in the whole space. We prove that the minimal and maximal discontinuous viscosity solutions of the associated Hamilton–Jacobi can be written in terms of value functions of control problems with reflection. Nethertheless, we construct a counter-example showing that the value function is not the unique solution of the equation. To cite this article: O. Ley, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 469–473.     

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the convexity of reachability sets of controlled initial-boundary value problems

For a functional-operator equation describing a broad class of controlled initial-boundary value problems, we introduce the notion of abstract reachability set. We obtain sufficient conditions for the convexity and precompactness of that set. The situation of a Nash ?-equilibrium is justified in the sense of program strategies in noncooperative functional-operator games with many players. As an example of reduction of a controlled initial-boundary value problem to the equation under study, we consider the Cauchy problem for a semilinear wave equation with two space variables.     

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L²(-p, p)\mathcal{L}^2(-\pi, \pi). Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.     

Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation

Based on a mathematical model of laser beams, we present a spectral Galerkin method for solving a Cauchy problem of the Helmholtz equation in a rectangle, where the Cauchy data pairs are given at y?=?0 and boundary data are for x?=?0 and x?=?π. The solution is sought in the interval 0?<?y?<?1. The spectral Galerkin method is considered as a regularization method. We then perform an analysis on the error bound for this method. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

On the relationship of some Fredholm integral equations of the first kind to a family of boundary value problems

It is shown that the solution of a class of forced convection heat transfer problems can be used as a vehicle for exhibiting a correspondence between certain boundary value problems and their associated integral equations. If the solution of the boundary value problem is known then the solution of the integral equation can be found by a simple calculation. This generalizes the relationship of certain solutions of Laplace's equation to integral equations having logarithmic or Cauchy kernels. When the boundary value problems have separable solutions, the solution of the integral equation can be reduced to the verification of a set of identities μ_n ∫_EP(t) φ_n(t) k(x ? t) dt = φ_n(x), x?E, where the {φ_n} form an orthonormal set on E.Once the method of solution has been derived from the physical approach it can be put on a firm mathematical basis.     

The genuinely nonlinear Graetz—Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in L _loc ¹ . The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar H-measures proposed by E. Yu. Panov.