scholarly journals WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

2021 ◽  
Vol 18 (03) ◽  
pp. 557-608
Author(s):  
Antoine Benoit
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Step Function ◽  
Geometric Optics ◽  
Loss Of Derivatives ◽  
Hyperbolic Boundary ◽  
Well Posed

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

scholarly journals Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

2011 ◽  
Vol 152 (3) ◽  
pp. 473-496 ◽  
Author(s):  
DAVID A. SMITH
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Transform Method ◽  
Arbitrary Boundary ◽  
Initial Boundary Value ◽  
Well Posed

AbstractWe study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

scholarly journals WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

2018 ◽  
Vol 19 (5) ◽  
pp. 1629-1675
Author(s):  
Antoine Benoit
Keyword(s):  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Well Posedness ◽  
Characteristic Variety ◽  
Strip Geometry ◽  
Initial Boundary Value ◽  
Hyperbolic Boundary ◽  
Wkb Expansion

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations

2003 ◽  
Author(s):  
H. N. Narang ◽  
Rajiv K. Nekkanti
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Initial Boundary ◽  
Time Dependent ◽  
Initial Boundary Value Problems ◽  
Non Linear ◽  
Initial Boundary Value

The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.

scholarly journals A MODEL PROBLEM FOR THE INITIAL-BOUNDARY VALUE FORMULATION OF EINSTEIN'S FIELD EQUATIONS

2005 ◽  
Vol 02 (02) ◽  
pp. 397-435 ◽  
Author(s):  
OSCAR REULA ◽  
OLIVIER SARBACH
Keyword(s):  
General Relativity ◽  
Boundary Conditions ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Einstein's Field Equations ◽  
Initial Boundary Value ◽  
Well Posed

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of space–time with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein–Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein–Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.

scholarly journals Biomechanics of the cardiovascular system: the aorta as an illustratory example

2006 ◽  
Vol 3 (11) ◽  
pp. 719-740 ◽  
Author(s):  
Ghassan S Kassab
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Material Properties ◽  
Vessel Wall ◽  
Boundary Value ◽  
Biomechanical Analysis ◽  
Surrounding Tissue ◽  
Balance Laws ◽  
Mechanical Homeostasis ◽  
Well Posed

Biomechanics relates the function of a physiological system to its structure. The objective of biomechanics is to deduce the function of a system from its geometry, material properties and boundary conditions based on the balance laws of mechanics (e.g. conservation of mass, momentum and energy). In the present review, we shall outline the general approach of biomechanics. As this is an enormously broad field, we shall consider a detailed biomechanical analysis of the aorta as an illustration. Specifically, we will consider the geometry and material properties of the aorta in conjunction with appropriate boundary conditions to formulate and solve several well-posed boundary value problems. Among other issues, we shall consider the effect of longitudinal pre-stretch and surrounding tissue on the mechanical status of the vessel wall. The solutions of the boundary value problems predict the presence of mechanical homeostasis in the vessel wall. The implications of mechanical homeostasis on growth, remodelling and postnatal development of the aorta are considered.

Sign in / Sign up

Export Citation Format

Share Document