Elliptic Part Research Articles

Basis property of the eigenfunctions of a generalized gasdynamic Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

We find the eigenfunctions of a generalized Frankl problem with the use of Bessel functions. We prove that these eigenfunctions form a Riesz basis in the space L 2(D +), where D + is the elliptic part of the domain. In addition, we prove the Riesz basis property of a trigonometric function system and the completeness of this system in the space L 2(0, π/2).

Existence of regular solutions to a class of parabolic systems in two space dimensions with critical growth behaviour

We consider parabolic systems $$u_{t} - {\rm div} \left( a(t, x, u, \nabla u)\right) + a_{0}(t, x, u, \nabla u) = 0$$ in two space dimensions with initial and Dirichlet boundary conditions. The elliptic part including a0 is derived from a potential with quadratic growth in ∇u and is coercive and monotone. The term a0 may grow quadratically in ∇u and satisfies a sign condition a0 · u ≥ −K. We prove the existence of a regular long time solution verifying a regularity criterion of Arkhipova. No smallness is assumed on the data.

Solution of a Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We write out the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation in the form of a series for the case in which the elliptic part of the domain is a half-strip and the boundary data are nonzero only on the characteristics in the hyperbolic parts of the domain. We obtain new results on the basis property, completeness, and minimality of the system of sines with discontinuous phase used in the series representation of the solution of the Gellerstedt problem. We prove the uniform convergence and justify the possibility of term-by-term differentiation of the series.

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in $${\mathbb{R}}^2$$

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as t → ∞ of solutions to a linear, parabolic system of equations with time-dependent coefficients in Ω × (0, ∞), where Ω is a bounded domain. On ∂ Ω × (0, ∞) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function κ (t). This includes in particular situations when the coefficients may take different values on different parts of Ω and the boundaries between them can move with t but stabilize as t → ∞. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if κ ∈ L1(0, ∞), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the completeness of eigenfunctions of the Frankl problem with the oddness condition and with the discontinuity of the gradient of solution

In this paper, we write eigenfunctions of the Frankl problem for the Lavrentev–Bitsadze equation with the nonlocal evenness condition and with the discontinuity of the derivative of the solution in the line of change of equation type. It is proved that the eigenfunctions generate a complete system in L 2(D +), where p≥2, and D +, the elliptic part of the domain.

Monotone schemes for a class of nonlinear elliptic and parabolic problems

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients satisfying the strict ellipticity condition, and that the right-hand side is a positive Radon measure. The numerical schemes are not derived from finite difference operators approximating differential operators, but rather from a general principle which ensures the convergence of approximate solutions. The main feature of these schemes is that they possess stencils stretching far from basic grid-rectangles, thus leading to system matrices which are related to M-matrices.

Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue

Uniqueness problems in the elliptic sector of constrained formulations of Einstein equations have a dramatic effect on the physical validity of some numerical solutions, for instance, when calculating the spacetime of very compact stars or nascent black holes. The fully constrained formulation (FCF) proposed by Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak is one of these formulations. It contains, as a particular case, the approximation of the conformal flatness condition (CFC) which, in the last ten years, has been used in many astrophysical applications. The elliptic part of the FCF basically shares the same differential operators as the elliptic equations in the CFC scheme. We present here a reformulation of the elliptic sector of the CFC that has the fundamental property of overcoming the local uniqueness problems. The correct behavior of our new formulation is confirmed by means of a battery of numerical simulations. Finally, we extend these ideas to the FCF, complementing the mathematical analysis carried out in previous studies.

Well-Posedness of the Cauchy Problem for Hyperbolic Equations with Non-Lipschitz Coefficients

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients. We prove well-posedness of the corresponding Cauchy problem in some functional spaces. These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothness with respect to variables corresponding to singular coefficients.

Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions

This paper studies a wave equation on a bounded domain in $\bbR^d$ with nonlinear dissipation which is localized on a subset of the boundary. The damping is modeled by a continuous monotone function without the usual growth restrictions imposed at the origin and infinity. Under the assumption that the observability inequality is satisfied by the solution of the associated <em>linear</em> problem, the asymptotic decay rates of the energy functional are obtained by reducing the <em>nonlinear PDE</em> problem to a <em> linear PDE</em> and a <em>nonlinear ODE</em>. This approach offers a generalized framework which incorporates the results on energy decay that appeared previously in the literature; the method accommodates systems with variable coefficients in the principal elliptic part, and allows to dispense with linear restrictions on the growth of the dissipative feedback map.

DECAY PROPERTY OF REGULARITY-LOSS TYPE AND NONLINEAR EFFECTS FOR SOME HYPERBOLIC-ELLIPTIC SYSTEM

We study the decay property of a certain nonlinear hyperbolic-elliptic system with 2mth-order elliptic part, which is a modified version of the simplest radiating gas model. It is proved that, for m ≥ 2, the system verifies a decay property of the regularity-loss type that is characterized by the parameter m. This dissipative property is very weak in the high-frequency region and causes a difficulty in showing the global existence of solutions to the nonlinear problem. By employing the time-weighted energy method together with the optimal decay for lower-order derivatives of solutions, we overcome this difficulty and establish a global existence and asymptotic decay result. Furthermore, we show that the solution approaches the nonlinear diffusion wave described by the self-similar solution of the Burgers equation as time tends to infinity.

Operational Space Calibration for Remote Welding Based on Surface Tracking With Shared Force Control

This paper presents a novel operational space calibration approach for robotic remote welding based on surface tracking with shared force control. A human-machine shared force controller is designed to combine manual control with local force control. A position-based force control strategy for surface tracking on the constrained motion plane is adopted. A precise method to measure the contact point’s spatial location during the surface tracking process is proposed. The operational space calibration for the L-pipe part is solved by direct least-squares fitting algorithm of elliptic tracking trajectory and L-pipe part calibration algorithm. The experimental results show that the proposed calibration method can make the operational space model error less than 1mm, which meets the requirements of carrying out assembling contact tasks during the remote welding process with passive compliance.

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition

The Frankl problem without the spectral parameter was considered by Bitsadze and Smirnov. The present paper gives the eigenvalues and eigenfunctions of the Frankl problem with the odd parity condition. We prove the completeness of eigenfunctions. The Frankl problem with a nonlocal parity condition for the Lavrent’ev-Bitsadze equation is studied. The eigenvalues and eigenfunctions are found, and the basis property of the eigenfunctions in the elliptic part of the domain in the space L 2 is proved.

Mathematical issues in a fully constrained formulation of the Einstein equations

Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D 70, 104007 (2004)] proposed a new formulation for $3+1$ numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular, focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole space-time evolutions.

On the completeness of eigenfunctions of the Frankl problem with the oddness condition

Modified Frankl problems with the oddness condition are considered for the Lavrent’ev-Bitsadze equation. The eigenvalues and eigenfunctions of these problems are found. The completeness of these eigenfunctions in the elliptic part of the domain is proved.

Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity

Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation). The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homo geneous elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities. The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/ sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute the fine-scale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of nonlinear compressible multiphase flow in strongly heterogeneous formations.

Decay and local eventual positivity for biharmonic parabolic equations

We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator $\partial_t+\Delta^2$ is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We determine conditions on the initial datum for which the corresponding solution exhibits some kind of positivity behaviour. We prove eventual local positivity properties both in the linear and semilinear case. At the same time, we show that negativity of the solution may occur also for arbitrarily large given time, provided the initial datum is suitably constructed.

Homogenization and correctors for the wave equation in non periodic perforated domains

We consider here the wave equation in a (not necessarily periodic) perforated domain, with a Neumann condition on the boundary of the holes. Assuming $H^0$-convergence ([3]) on the elliptic part of the operator, we prove two main theorems: a convergence result and a corrector one. To prove the corrector result, we make use of a suitable family of elliptic local correctors given in [4] whose columns are piecewise locally square integrable gradients. As in the case without holes ([2]), some additional assumptions on the data are needed.

Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system.

Diffractive wave transmission in dispersive media

The aim of this paper is to study the reflection–transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarization. The framework is that of P. Donnat's thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485–535]: we consider an infinite WKB expansion of the wave over long times/distances O ( 1 / ε ) and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132–209]. Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider E , the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on E which leads us to make a restriction on the set of resonant phases. We finally give a convergence result which justifies the use of “quasi compactly” supported profiles.