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Discrete & Continuous Dynamical Systems - A

2009 , Volume 23 , Issue 1/2

A special issue
Dedicated to Ta-Tsien Li on the Occasion of his 70th Birthday

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Preface
Philippe G. Ciarlet
2009, 23(1/2): i-ii doi: 10.3934/dcds.2009.23.1i +[Abstract](78) +[PDF](31.2KB)
Abstract:
Ta-Tsien Li is since 1980 Professor at the prestigious Fudan University in Shanghai, one of the best universities in China. He began his scientific career as a student of the famous mathematician Chao-hao Gu, also at Fudan University. After he finished his in-service graduate study in 1966, he spent the difficult years of the so-called “Cultural" Revolution in total isolation. It is only after 1976 that he could begin to resume the usual scientific activities. In this respect, the two years that he spent from 1979-1981 as a Research Fellow at the celebrated Collège de France in Paris were decisive. It is the late Professor Jacques-Louis Lions, one of the most eminent and influential applied mathematicicians of the twentieth century, who had invited Ta-Tsien Li in Paris, a sure sign that he had an excellent opinion of him! There he became acquainted with the theory of partial differential equations and control theory, together with some of their manifold applications, such as nonlinear elasticity or gas dynamics.

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Foreword
Gui-Qiang Chen and Jiaxing Hong
2009, 23(1/2): iii-vi doi: 10.3934/dcds.2009.23.1iii +[Abstract](78) +[PDF](54.4KB)
Abstract:
Professor Ta-Tsien Li was born on November 10, 1937, in Nantong, Jiangsu Province, China. He was graduated in 1957 from the Department of Mathematics, Fudan University, and has been its faculty member since then. His in-service graduate study as a student of Professor Chao-Hao Gu at the university finished in 1966. At the invitation of Professor Jacques-Louis Lions, Professor Li visited the prestigious Collège de France, Paris, France, as a visiting scholar from January 1979 to April 1981. He was promoted to be a Full Professor of Mathematics in 1980, became a Ph.D. Supervisor for Pure Mathematics in 1981 and Applied Mathematics in 1983 respectively, and was appointed as the Dean of Graduate School of Fudan University from 1991 to 1999. Professor Li was elected as a Member of the Chinese Academy of Sciences in 1995, a Fellow of the Third World Academy of Sciences (the Academy of Sciences for the Developing World) in 1997, a Foreign Member of the French Academy of Sciences in 2005 and a Member of the European Academy of Sciences in 2007.

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The convergence of the GRP scheme
Matania Ben–Artzi, Joseph Falcovitz and Jiequan Li
2009, 23(1/2): 1-27 doi: 10.3934/dcds.2009.23.1 +[Abstract](59) +[PDF](342.3KB)
Abstract:
This paper deals with the convergence of the second-order GRP (Generalized Riemann Problem) numerical scheme to the entropy solution for scalar conservation laws with strictly convex fluxes. The approximate profiles at each time step are linear in each cell, with possible jump discontinuities (of functional values and slopes) across cell boundaries. The basic observation is that the discrete values produced by the scheme are exact averages of an approximate conservation law , which enables the use of properties of such solutions in the proof. In particular, the “total-variation" of the scheme can be controlled, using analytic properties. In practice, the GRP code allows “sawteeth" profiles (i.e., the piecewise linear approximation is not monotone even if the sequences of averages is such). The “reconstruction" procedure considered here also allows the formation of “sawteeth" profiles, with an hypothesis of “Godunov Compatibility", which limits the slopes in cases of non-monotone profiles. The scheme is proved to converge to a weak solution of the conservation law. In the case of a monotone initial profile it is shown (under a further hypothesis on the slopes) that the limit solution is indeed the entropy solution. The constructed solution satisfies the “finite propagation speed", so that no rarefaction shocks can appear in intervals such that the initial function is monotone in their domain of dependence. However, the characterization of the limit solution as the unique entropy solution, for general initial data, is still an open problem.
On the convergence of viscous approximations after shock interactions
Alberto Bressan and Carlotta Donadello
2009, 23(1/2): 29-48 doi: 10.3934/dcds.2009.23.29 +[Abstract](80) +[PDF](221.8KB)
Abstract:
We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient.
Nonlocal heat flows preserving the L2 energy
Luis Caffarelli and Fanghua Lin
2009, 23(1/2): 49-64 doi: 10.3934/dcds.2009.23.49 +[Abstract](109) +[PDF](215.9KB)
Abstract:
We shall study L2 energy conserved solutions to the heat equation. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a family of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.
Robust filtering for joint state-parameter estimation in distributed mechanical systems
Dominique Chapelle, Philippe Moireau and Patrick Le Tallec
2009, 23(1/2): 65-84 doi: 10.3934/dcds.2009.23.65 +[Abstract](83) +[PDF](1170.3KB)
Abstract:
We present an effective filtering procedure for jointly estimating state variables and parameters in a distributed mechanical system. This method is based on a robust, low-cost filter related to collocated feedback and used to estimate state variables, and an H setting is then employed to formulate a joint state-parameter estimation filter. In addition to providing a tractable filtering approach for an infinite-dimensional mechanical system, the H setting allows to consider measurement errors that cannot be handled by Kalman type filters, e.g. for measurements only available on the boundary. For this estimation strategy a complete error analysis is given, and a detailed numerical assessment -- using a test problem inspired from cardiac biomechanics -- demonstrates the effectiveness of our approach.
Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone
Gui-Qiang Chen and Beixiang Fang
2009, 23(1/2): 85-114 doi: 10.3934/dcds.2009.23.85 +[Abstract](67) +[PDF](422.2KB)
Abstract:
For an upstream supersonic flow past a straight-sided cone in R3 whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.
Weak shock solution in supersonic flow past a wedge
Shuxing Chen, Jianzhong Min and Yongqian Zhang
2009, 23(1/2): 115-132 doi: 10.3934/dcds.2009.23.115 +[Abstract](99) +[PDF](206.2KB)
Abstract:
In this paper we study the local existence and uniqueness of weak shock solution in steady supersonic flow past a wedge. We take the 3-D potential flow equation as the mathematical model to describe the compressible flow. It is known that when a supersonic flow passes a wedge, there will appear an attached shock front, provided that the vertex angle of the wedge is less than a critical value. In generic case the problem admits two possible locations of the shock front, connecting the flow ahead of it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). In this paper we prove the local existence and uniqueness of weak shock front if the coming flow is a small perturbation of a constant supersonic flow. Our analysis is based on the usage of partial hodograph transformation and domain decomposition, which let the proof be simpler than the previous discussion.
Intrinsic methods in elasticity: a mathematical survey
Philippe G. Ciarlet, Liliana Gratie and Cristinel Mardare
2009, 23(1/2): 133-164 doi: 10.3934/dcds.2009.23.133 +[Abstract](75) +[PDF](389.2KB)
Abstract:
In the classical approach to elasticity problems, the components of the displacement field are the primary unknowns. In an "intrinsic'' approach, new unknowns with more physical or geometrical meanings, such as a strain tensor field or a rotation field for instance, are instead taken as the primary unknowns. We survey here recent progress about the mathematical analysis of such methods applied to linear and nonlinear three-dimensional elasticity and shell problems.
Levelsets and anisotropic mesh adaptation
Alexandra Claisse, Pascal Frey and Vincent Ducrot
2009, 23(1/2): 165-183 doi: 10.3934/dcds.2009.23.165 +[Abstract](56) +[PDF](732.5KB)
Abstract:
In this paper, we focus on the problem of adapting dynamic triangulations during numerical simulations to reduce the approximation errors. Dynamically evolving interfaces arise in many applications, such as free surfaces in multiphase flows and moving surfaces in fluid-structure interactions. In such simulations, it is often required to preserve a high quality interface discretization thus posing significant challenges in adapting the triangulation in the vicinity of the interface, especially if its geometry or its topology changes dramatically during the simulation. Our approach combines an efficient levelset formulation to represent the interface in the flow equations with an anisotropic mesh adaptation scheme based on a Riemannian metric tensor to prescribe size, shape and orientation of the elements. Experimental results are provided to emphasize the effectiveness of this technique for dynamically evolving interfaces in flow simulations.
A variational approach to the Riemann problem for hyperbolic conservation laws
Constantine M. Dafermos
2009, 23(1/2): 185-195 doi: 10.3934/dcds.2009.23.185 +[Abstract](84) +[PDF](152.6KB)
Abstract:
Within the framework of strictly hyperbolic systems of conservation laws endowed with a convex entropy, it is shown that the admissible solution to the Riemann problem is obtained by minimizing the entropy production over all wave fans with fixed end-states.
Homogenization of oscillating boundaries
Klas Pettersson and Alain Damlamian
2009, 23(1/2): 197-219 doi: 10.3934/dcds.2009.23.197 +[Abstract](57) +[PDF](264.5KB)
Abstract:
A variational problem on a sequence of 2-dimensional domains with oscillating boundaries is studied. Using the periodic unfolding method, the homogenized problem is obtained in the limit as the period length approaches zero. Several extensions are also given. In this framework, a result of strong convergence is obtained which is new.
Third order equivalent equation of lattice Boltzmann scheme
François Dubois
2009, 23(1/2): 221-248 doi: 10.3934/dcds.2009.23.221 +[Abstract](107) +[PDF](307.6KB)
Abstract:
We recall the origin of lattice Boltzmann scheme and detail the version due to D'Humières [8]. We present a formal analysis of this lattice Boltzmann scheme in terms of a single numerical infinitesimal parameter. We derive third order equivalent partial differential equation of this scheme. Both situations of single conservation law and fluid flow with mass and momentum conservations are detailed. We apply our analysis to so-called D1Q3 and D2Q9 lattice Boltzmann schemes in one and two space dimensions.
Orbital minimization with localization
Weiguo Gao and Weinan E
2009, 23(1/2): 249-264 doi: 10.3934/dcds.2009.23.249 +[Abstract](83) +[PDF](1090.9KB)
Abstract:
Orbital minimization is among the most promising linear scaling algorithms for electronic structure calculation. However, to achieve linear scaling, one has to truncate the support of the orbitals and this introduces many problems, the most important of which is the occurrence of numerous local minima. In this paper, we introduce a simple modification of the orbital minimization method, by adding a localization step into the algorithm. This localization step selects the most localized representation of the subspace spanned by the orbitals obtained during the intermediate stages of the iteration process. We show that the addition of the localization step substantially reduces the chances that the iterations get trapped at local minima.
Numerical zoom for multiscale problems with an application to flows through porous media
Frédéric Hecht, Alain Perronnet, Alexei Lozinski and Olivier Pironneau
2009, 23(1/2): 265-280 doi: 10.3934/dcds.2009.23.265 +[Abstract](114) +[PDF](847.4KB)
Abstract:
We propose a technique for interactive mesh refinement in regions where the solution of a partial differential equation is less regular. Based on the method of harmonic patches, the idea is to bypass an expensive calculation on a fine mesh and yet retain the same accuracy with several much smaller computations. A general numerical zoom method is presented; then it is specialized to the case where the mesh in the zoom is a refinement of the coarse mesh; it is also compared with classic domain decomposition algorithms. Numerical examples are given for a porous flow modeled by Darcy's law.
Multiscale analysis for convection dominated transport equations
Dong Liang and Thomas Y. Hou
2009, 23(1/2): 281-298 doi: 10.3934/dcds.2009.23.281 +[Abstract](89) +[PDF](246.8KB)
Abstract:
In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on upscaling linear transport equations. But we also discuss briefly how to upscale two-phase miscible flows, in which case the concentration equation is coupled to the pressure equation in a nonlinear fashion. For the problem we consider here, the local Peclet number is of order $O(\epsilon^{-m+1})$ with $m \in [2,\infty]$ being any integer, where $\epsilon$ characterizes the small scale in the heterogeneous media. Due to the presence of the nonlocal memory effect, upscaling a convection dominated transport equation is known to be very difficult. One of the key ideas in deriving a well-posed homogenized equation for the convection dominated transport equation is to introduce a projection operator which projects the fluctuation onto a suitable subspace. This projection operator corresponds to averaging along the streamlines of the flow. In the case of linear convection dominated transport equations, we prove the well-posedness of the homogenized equations and establish rigorous error estimates for our multiscale expansion.
On global controllability of 2-D Burgers equation
Jean Pierre Puel and Oleg Yu. Imanuvilov
2009, 23(1/2): 299-313 doi: 10.3934/dcds.2009.23.299 +[Abstract](99) +[PDF](200.4KB)
Abstract:
We study the question of global controllability for the two-dimensional Burgers equation when the control acts on a part $\Gamma_{1}$ of the boundary $\Gamma$. We prove global controllability when $\Gamma_{1}$ is the whole boundary or in a specific geometrical situation when $\Gamma_{0}=\Gamma \setminus \Gamma_{1}$ is contained in a parallel to the first bisector line. We also show with a counterexample that $\Gamma_{1}$ cannot be taken any part of the boundary.
Interaction of boundary layers and corner singularities
Chang-Yeol Jung and Roger Temam
2009, 23(1/2): 315-339 doi: 10.3934/dcds.2009.23.315 +[Abstract](68) +[PDF](9194.2KB)
Abstract:
Our aim in this article is to study the interaction of \textit{boundary layers} and \textit{corner singularities} in the context of singularly perturbed convection-diffusion equations. For the problems under consideration, we determine a simplified form of the corner singularities and show how to use it for the numerical approximation of such problems in the context of variational approximations using the concept of \textit{enriched spaces}.
Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity
Philippe G. Lefloch, Cristinel Mardare and Sorin Mardare
2009, 23(1/2): 341-365 doi: 10.3934/dcds.2009.23.341 +[Abstract](82) +[PDF](296.6KB)
Abstract:
Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changes from point to point.
A Carleman estimate for the linear shallow shell equation and an inverse source problem
Bernadette Miara, Masahiro Yamamoto and Shumin Li
2009, 23(1/2): 367-380 doi: 10.3934/dcds.2009.23.367 +[Abstract](89) +[PDF](187.4KB)
Abstract:
We consider an elastic bi-dimensional body whose reference configuration is a shallow shell. We establish a Carleman estimate for the linear shallow shell equations and apply it to prove a conditional stability for an inverse problem of determining external source terms by observations of displacement in a neighbourhood of the boundary over a time interval.
The initial-boundary value problem on a strip for the equation of time-like extremal surfaces
Jianli Liu and Yi Zhou
2009, 23(1/2): 381-397 doi: 10.3934/dcds.2009.23.381 +[Abstract](104) +[PDF](218.6KB)
Abstract:
In this paper we study the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ on the strip $R^{+}\times[0,1]$. Under the assumptions that the boundary data are small and decaying, we get the global existence and uniqueness of classical solutions.
A spectral approach to the indirect boundary control of a system of weakly coupled wave equations
Zhuangyi Liu and Bopeng Rao
2009, 23(1/2): 399-414 doi: 10.3934/dcds.2009.23.399 +[Abstract](89) +[PDF](221.2KB)
Abstract:
In this paper, we study the exact controllability of a system of two weakly coupled one-dimensional wave equations with the control acted on only one equation. Using the non harmonic analysis, we establish the weak observability inequalities, which depend on the ratio of the wave propagation speeds. The obtained results are optimal.
Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters
Shu Wang and Yue-Jun Peng
2009, 23(1/2): 415-433 doi: 10.3934/dcds.2009.23.415 +[Abstract](99) +[PDF](256.4KB)
Abstract:
This work is concerned with the two-fluid Euler-Maxwell equations for plasmas with small parameters. We study, by means of asymptotic expansions, the zero-relaxation limit, the non-relativistic limit and the combined non-relativistic and quasi-neutral limit. For each limit with well-prepared initial data, we show the existence and uniqueness of an asymptotic expansion up to any order. For general data, an asymptotic expansion up to order 1 of the non-relativistic limit is constructed by taking into account the initial layers. Finally, we discuss the justification of the limits.
Non-linear electromagnetism and special relativity
Denis Serre
2009, 23(1/2): 435-454 doi: 10.3934/dcds.2009.23.435 +[Abstract](64) +[PDF](241.8KB)
Abstract:
We continue the study of nonlinear Maxwell equations for electromagnetism in the formalism of B. D. Coleman & E. H. Dill. We exploit here the assumption of Lorentz invariance, following I. Białinicki-Barula. In particular, we show that nonlinearity forbids the convexity of the electromagnetic energy density. This justifies the study of rank-one convex and of polyconvex densities, begun in [8, 16]. We also show the alternative that either electrodynamics is linear, or dispersion is lost as the electromagnetic field becomes intense.
On energetic variational approaches in modeling the nematic liquid crystal flows
Huan Sun and Chun Liu
2009, 23(1/2): 455-475 doi: 10.3934/dcds.2009.23.455 +[Abstract](140) +[PDF](237.6KB)
Abstract:
In this paper we present results for the existence of classical solutions of a hydrodynamical system modeling the flow of nematic liquid crystals. The system consists of a coupled system of Navier-Stokes equations and various kinematic transport equations for the molecular orientations. A formal physical derivation of the induced elastic stress using least action principle reflects the special coupling between the transport and the induced stress terms. The derivation and the analysis of the system falls into a general energetic variational framework for complex fluids with elastic effects due to the presence of nontrivial microstructures.
Quasilinear elliptic equations with signed measure
Xu-Jia Wang and Neil S. Trudinger
2009, 23(1/2): 477-494 doi: 10.3934/dcds.2009.23.477 +[Abstract](257) +[PDF](251.3KB)
Abstract:
This paper treats quasilinear elliptic equations in divergence form whose inhomogeneous term is a signed measure. We first prove the existence and continuity of generalized solutions to the Dirichlet problem. The main result of this paper is a weak convergence result, extending previous work of the authors for subharmonic functions and non-negative measures. We also prove a uniqueness result for uniformly elliptic operators and for operators of $p$-Laplacian type.
Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence
Huijiang Zhao, Seiji Ukai and Tong Yang
2009, 23(1/2): 495-520 doi: 10.3934/dcds.2009.23.495 +[Abstract](67) +[PDF](331.5KB)
Abstract:
The exterior problem arising from the study of a flow past an obstacle is one of the most classical and important subjects in gas dynamics and fluid mechanics. The point of this problem is to assign the bulk velocity at infinity, which is not a trivial driving force on the flow so that some non-trivial solution profiles persist. In this paper, we consider the exterior problem for the Boltzmann equation when the Mach number of the far field equilibrium state is small. The result here generalizes the previous one by Ukai-Asano on the same problem to more general boundary conditions by crucially using the velocity average argument.
Upper semi-continuity of stationary statistical properties of dissipative systems
Xiaoming Wang
2009, 23(1/2): 521-540 doi: 10.3934/dcds.2009.23.521 +[Abstract](65) +[PDF](234.4KB)
Abstract:
We show that stationary statistical properties for uniformly dissipative dynamical systems are upper semi-continuous under regular perturbation and a special type of singular perturbation in time of relaxation type. The results presented are applicable to many physical systems such as the singular limit of infinite Prandtl-Darcy number in the Darcy-Boussinesq system for convection in porous media, or the large Prandtl asymptotics for the Boussinesq system.
On the connection formulas of the third Painlevé transcendent
Roderick S. C. Wong and H. Y. Zhang
2009, 23(1/2): 541-560 doi: 10.3934/dcds.2009.23.541 +[Abstract](86) +[PDF](252.2KB)
Abstract:
We consider the connection problem for the sine-Gordon PIII equation $u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general third Painlevé transcendents. The connection formulas are derived by the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. Anal., 1998).
Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves
Huaxiong Huang, Robert M. Miura and Jonathan J. Wylie
2009, 23(1/2): 561-569 doi: 10.3934/dcds.2009.23.561 +[Abstract](76) +[PDF](138.1KB)
Abstract:
Diffusion equations with degenerate nonlinear source terms arise in many different applications, e.g., in the theory of epidemics, in models of cortical spreading depression, and in models of evaporation and condensation in porous media. In this paper, we consider a generalization of these models to a system of $n$ coupled diffusion equations with identical nonlinear source terms. We determine simple conditions that ensure the linear stability of uniform rest states and show that traveling wave trajectories connecting two stable rest states can exist generically only for discrete wave speeds. Furthermore, we show that families of traveling waves with a continuum of wave speeds cannot exist.
Time discrete wave equations: Boundary observability and control
Xu Zhang, Chuang Zheng and Enrique Zuazua
2009, 23(1/2): 571-604 doi: 10.3934/dcds.2009.23.571 +[Abstract](98) +[PDF](932.3KB)
Abstract:
In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step $h\rightarrow 0$, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to $h\rightarrow 0$) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.
Absorption of characteristics by sonic curve of the two-dimensional Euler equations
Yuxi Zheng
2009, 23(1/2): 605-616 doi: 10.3934/dcds.2009.23.605 +[Abstract](62) +[PDF](194.9KB)
Abstract:
We explore the reflection off a sonic curve and the domain of determinacy, via the method of characteristics, of self-similar solutions to the two dimensional isentropic Euler system through several examples with axially symmetric initial data. We find that characteristics in some cases can be completely absorbed by the sonic curve so that the characteristics vanish tangentially into the sonic boundary, exemplifying a classical scenario of the Keldysh type; however, the characteristics can wrap around the closed sonic curve unboundedly many times, so that the domain of determinacy of the hyperbolic characteristic boundary value problem or the Goursat problem exhibit layered annulus structures. As the number of layers increases, the layers become thinner, and the solution at an interior point of the domain depends eventually on the entire boundary data.

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