System Of Partial Dierential Equations Research Articles

The onset of instability in first-order systems

We study the Cauchy problem for rst-order quasi-linear systems of partial dierential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linear Lax-Mizohata theorem, was explained by G. M etivier in ( Remarks on the well-posedness of the nonlinear Cauchy problem, Contemp. Math. 2005). In the present paper, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under such an hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto and C.-J. Xu (Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems, American J. Math. 2010) on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein-Gordon-Zakharov systems. Our analysis relies on an approximation result for pseudo-dierentia l ows, introduced by B. Texier in (Approximations of pseudo-dierentia l ows , arxiv:1402.6868).

LED-Based Photometric Stereo: Modeling, Calibration and Numerical Solution

We conduct a thorough study of photomet-ric stereo under nearby point light source illumination, from modeling to numerical solution, through calibration. In the classical formulation of photometric stereo, the luminous uxes are assumed to be directional, which is very dicult to achieve in practice. Rather, we use light-emitting diodes (LEDs) to illuminate the scene to be reconstructed. Such point light sources are very convenient to use, yet they yield a more complex photo-metric stereo model which is arduous to solve. We rst derive in a physically sound manner this model, and show how to calibrate its parameters. Then, we discuss two state-of-the-art numerical solutions. The rst one alternatingly estimates the albedo and the normals, and then integrates the normals into a depth map. It is shown empirically to be independent from the ini-tialization, but convergence of this sequential approach is not established. The second one directly recovers the depth, by formulating photometric stereo as a system of nonlinear partial dierential equations (PDEs), which are linearized using image ratios. Although the sequential approach is avoided, initialization matters a lot and convergence is not established either. Therefore, we introduce a provably convergent alternating reweighted least-squares scheme for solving the original system of nonlinear PDEs. Finally, we extend this study to the case of RGB images.

Multiscale Analysis of a Carcinogenesis Model

Using the periodic unfolding method, we shall analyze the macroscopic behavior of a non-linear system of partial dierential equations describing some biochemical processes involved in the carcinogenesis in human cells. We shall mainly focus here on some important phenomena arising in the carcinogenesis of living cells produced by Benzo[a]pyrene molecules.Various biochemical mechanisms, such as chemical reactions and diffusion processes occurring at the surface of the endoplasmic reticulum or binding and cleaning processes, described by suitable nonlinear functions, are taken into account in our model.

Wronskian and Grammian solutions to the (3+1)-dimensional BKP equation

In this paper, the (3+1)-dimensional BKP equation is mainly being discussed. Based on the the Wronskian technique and the Pfaan properties, Wronskian and Grammian solutions have been established. Generating functions for matrix entries satisfy linear system of partial dierential equations involving a free parameter.

The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity

The aim of this paper is to derive a formula for the solution to the Cauchy problem for the linear system of partial dierential equations

Conservation Laws of Discrete Korteweg-de Vries Equation

All three-point and five-point conservation laws for the discrete Korteweg-de Vries equations are found. These conservation laws satisfy a functional equation, which we solve by reducing it to a system of partial dierential equations. Our method uses computer algebra intensively, because the determining functional equation is quite complicated.

AQ-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system

The goal of this paper is to construct a rst-order upwind scheme for solving the system of partial dierential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Berm udez and V azquez-Cend on (3, 26, 27) for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The diculty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to dene a suitable numerical scheme with global upwinding, we rst consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.

Cauchy problem for hyperbolic 𝒟-modules with regular singularities

LetM be a cgoherentD-module (e.g., an overdetermined system of partial dierential equations) on the complexification of a real analytic manifold M. Assume that the characteristic variety ofM is hyperbolic with respect to a submanifold N M. Then, it is well-known that the Cauchy problem forM with data on N is well posed in the space of hyperfunctions. In this paper, under the additional assumption thatM has regular singularities along a regular involutive submanifold of real type, we prove that the Cauchy problem is well posed in the space of distributions. When M is induced by a single dierential operator (or by a normal square system) with characteristics of constant multiplicities, our hypotheses correspond to Levi conditions, and we recover a classical result.

On the boundary value problem in a dihedral angle for normally hyperbolic systems of first order

Some structural properties as well as a general three- dimensional boundary value problem for normally hyperbolic systems of partial dierential equations of Þrst order are studied. A condition is given which enables one to reduce the system under consideration to a Þrst-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulÞlled.

On Three-Dimensional Dynamic Problems of Generalized Thermoelasticity

Lord-Shulman's system of partial dierential equations of generalized thermoelasticity (1) is considered, in which the Þnite velocity of heat propagation is taken into account by introducing a relaxation time constant. General aspects of the theory of boundary value and initial-boundary value problems and representation of so- lutions by series and quadratures are considered using the method of a potential. As one knows, in the classical theory of thermoelasticity the velocity of heat propagation is assumed to be inÞnitely large. However, in study- ing dynamic thermal stresses in deformable solid bodies, when the inertia terms in the equations of motion cannot be neglected, one must take into account that heat propagates not with an inÞnite but with a Þnite velocity; a heat sow arises in the body not instantly but is characterized by a Þnite relaxation time. Presently, there are at least two dierent generalizations of the classical theory of thermoelasticity: the Þrst of them, Green-Lindsay's generalization (1) is based on using two heat relaxation time constants; the other one, Lord-Shulman's generalization (2) admits only one relaxation time constant. Both generalizations were developed as an attempt at ex- plaining the paradox of the classical case that the heat propagation velocity is an inÞnite value. In this paper, based on (3), we develop a general theory of solvability, as well as of construction of approximate and eective solutions of dynamic problems for the conjugate system of dierential equations of thermoelastic- ity proposed by Lord and Shulman (LAS theory). Green-Lindsay's theory (G A L-theory) is developed in (4, 5).

Nonstationary Initial Boundary Value Contact Problems of Generalized Elastothermodiffusion

The initial boundary value problems with mixed bound- ary conditions are considered for a system of partial dierential equa- tions of generalized electrothermodiusion. Approximate solutions are constructed and a mathematical substantiation of the method is given. It is well known that a wide range of problems of mathematical physics includes boundary value and initial boundary value problems for dieren- tial equations. Such problems are rather dicult to solve because of an enormous variety of geometrical forms of the investigated objects and the complexity of boundary conditions. Hence an important and timely task has arisen to develop suciently eective methods and tools for solving the above-mentioned problems and obtaining their numerical solutions. Until recently no methods were known for solving problems of elasticity by means of conjugate Þelds. However, in the past few years there have appeared and keep on appearing numerous published works dedicated to this topic. The results of our studies in this direction are presented mainly in (1, 2, 3). In this paper, using the generalized Green-Lindsay theory of elastother- modiusion as an example, the approaches to the solution of nonstationary initial boundary value contact problems with mixed boundary conditions are described for a system of partial dierential equations of this theory in a nonhomogeneous medium. These approaches are based on the method of discrete singularities known as the Riesz-Fischer-Kupradze method. The particular case is treated in (4).

On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial dierential equations in two independent variables which are integrated in quadratures. As well as developing approximate and numerical methods, finding exact solutions of the Euler equations (EEs) for an ideal incompressible fluid is an important problem of modern mathematical physics and hydrodynamics. There exist some ways to solve this problem. One of them is to use symmetry analysis [1, 2, 3]. We construct a complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the EEs. Using some of them, we reduce the EEs to systems of partial dierential equations in two independent variables which can be integrated in quadratures. As a result, we obtain classes of exact solutions of the EEs that contain arbitrary functions. It is known [4], that the EEs ~t + (~ · ~

On Lie Reduction of the Navier-Stokes Equations

Lie reduction of the Navier-Stokes equations to systems of partial dierential equations in three and two independent variables and to ordinary dierential equations is described.