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1078-0947

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1553-5231

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## Discrete & Continuous Dynamical Systems

January & February 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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2009, 23(1&2): i-ii
doi: 10.3934/dcds.2009.23.1i

*+*[Abstract](2119)*+*[PDF](31.2KB)**Abstract:**

Ta-Tsien Li is since 1980 Professor at the prestigious FudanUniversity in Shanghai, one of the best universities in China. Hebegan his scientific career as a student of the famous mathematician Chao-hao Gu, also at Fudan University. Afterhe finished his in-service graduate study in 1966, he spent thedifficult years of the so-called “Cultural" Revolution in totalisolation. It is only after 1976 that he could begin to resume theusual scientific activities. In this respect, the two years that hespent from 1979-1981 as a Research Fellow at the celebratedCollège de France in Paris were decisive. It is the late ProfessorJacques-Louis Lions, one of the most eminent and influential appliedmathematicicians of the twentieth century, who had invited Ta-TsienLi in Paris, a sure sign that he had an excellent opinion of him!There he became acquainted with the theory of partial differentialequations and control theory, together with some of their manifoldapplications, such as nonlinear elasticity or gas dynamics.

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2009, 23(1&2): iii-vi
doi: 10.3934/dcds.2009.23.1iii

*+*[Abstract](1861)*+*[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 theDepartment of Mathematics, Fudan University, and has been itsfaculty member since then. His in-service graduate study as astudent of Professor Chao-Hao Gu at the university finished in 1966.At the invitation of Professor Jacques-Louis Lions, Professor Livisited the prestigious Collège de France, Paris, France, as avisiting scholar from January 1979 to April 1981. He was promoted tobe a Full Professor of Mathematics in 1980, became a Ph.D.Supervisor for Pure Mathematics in 1981 and Applied Mathematics in1983 respectively, and was appointed as the Dean of Graduate Schoolof Fudan University from 1991 to 1999. Professor Li was elected as aMember of the Chinese Academy of Sciences in 1995, a Fellow of theThird World Academy of Sciences (the Academy of Sciences for theDeveloping World) in 1997, a Foreign Member of the French Academy ofSciences in 2005 and a Member of the European Academy of Sciences in2007.

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2009, 23(1&2): 1-27
doi: 10.3934/dcds.2009.23.1

*+*[Abstract](2286)*+*[PDF](342.3KB)**Abstract:**

This paper deals with the convergence of the second-order GRP(Generalized Riemann Problem) numerical scheme to the entropysolution 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 discretevalues produced by the scheme are

**exact averages**of an

**approximate conservation law**, which enables the use of propertiesof such solutions in the proof. In particular, the“total-variation" of the scheme can be controlled, using analyticproperties. In practice, the GRP code allows “sawteeth" profiles(i.e., the piecewise linear approximation is not monotone even ifthe sequences of averages is such). The “reconstruction" procedureconsidered here also allows the formation of “sawteeth" profiles,with an hypothesis of “Godunov Compatibility", which limits theslopes in cases of non-monotone profiles. The scheme is proved toconverge to a weak solution of the conservation law. In the case ofa monotone initial profile it is shown (under a further hypothesison the slopes) that the limit solution is indeed the entropysolution. The constructed solution satisfies the “finitepropagation speed", so that no rarefaction shocks can appear inintervals such that the initial function is monotone in their domainof dependence. However, the characterization of the limit solutionas the unique entropy solution, for general initial data, is stillan open problem.

2009, 23(1&2): 29-48
doi: 10.3934/dcds.2009.23.29

*+*[Abstract](2157)*+*[PDF](221.8KB)**Abstract:**

We considera piecewise smooth solution to a scalar conservation law,with possiblyinteracting shocks.We show that, after the interactions have taken place,vanishing viscosity approximations can still be represented bya regular expansion on smooth regions and by asingular perturbation expansion near the shocks, in termsof powers of the viscosity coefficient.

2009, 23(1&2): 49-64
doi: 10.3934/dcds.2009.23.49

*+*[Abstract](2639)*+*[PDF](215.9KB)**Abstract:**

We shall study

*L*

^{2}energy conserved solutions to the heat equation.We shall first establish the global existence, uniqueness andregularity of solutions to such nonlocal heat flows. We then extend themethod to a family of singularly perturbed systems of nonlocal parabolicequations. The main goal is to show that solutions to these perturbedsystems converges strongly to some suitable weak-solutionsof the limiting constrained nonlocal heat flows of maps into a singularspace. It is then possible to study further properties of such suitableweak solutions and the corresponding free boundary problem, which willbe discussed in a forthcoming article.

2009, 23(1&2): 65-84
doi: 10.3934/dcds.2009.23.65

*+*[Abstract](2783)*+*[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.

2009, 23(1&2): 85-114
doi: 10.3934/dcds.2009.23.85

*+*[Abstract](2396)*+*[PDF](422.2KB)**Abstract:**

For an upstream supersonic flow past a straight-sided cone in

*R*

^{3}whose vertex angle is less than the critical angle, a transonic(supersonic-subsonic) shock-front attached to the cone vertex can beformed in the flow. In this paper we analyze the stability oftransonic shock-fronts in three-dimensional steady potential flowpast a perturbed cone. We establish that the self-similar transonicshock-front solution is conditionally stable in structure withrespect to the conical perturbation of the cone boundary and theupstream flow in appropriate function spaces. In particular, it isproved that the slope of the shock-front tends asymptotically to theslope of the unperturbed self-similar shock-front downstream atinfinity.

2009, 23(1&2): 115-132
doi: 10.3934/dcds.2009.23.115

*+*[Abstract](2351)*+*[PDF](206.2KB)**Abstract:**

In this paper we study the local existence and uniqueness of weakshock solution in steady supersonic flow past a wedge. We take the 3-D potential flow equation as the mathematical model todescribe the compressible flow. It is known that when a supersonicflow passes a wedge, there will appear an attached shock front,provided that the vertex angle of the wedge is less than a criticalvalue. In generic case the problem admits two possible locations ofthe shock front, connecting the flow ahead of it and behind it. They can bedistinguished as supersonic-supersonic shock and supersonic-subsonicshock (or transonic shock). In this paper we prove the localexistence and uniqueness of weak shock front if the coming flow is asmall perturbation of a constant supersonic flow. Our analysis isbased on the usage of partial hodograph transformation and domaindecomposition, which let the proof be simpler than the previousdiscussion.

2009, 23(1&2): 133-164
doi: 10.3934/dcds.2009.23.133

*+*[Abstract](2560)*+*[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.

2009, 23(1&2): 165-183
doi: 10.3934/dcds.2009.23.165

*+*[Abstract](2376)*+*[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.

2009, 23(1&2): 185-195
doi: 10.3934/dcds.2009.23.185

*+*[Abstract](2322)*+*[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.

2009, 23(1&2): 197-219
doi: 10.3934/dcds.2009.23.197

*+*[Abstract](2188)*+*[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.

2009, 23(1&2): 221-248
doi: 10.3934/dcds.2009.23.221

*+*[Abstract](2985)*+*[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 analysisof this lattice Boltzmann scheme in terms of a single numericalinfinitesimal 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.

2009, 23(1&2): 249-264
doi: 10.3934/dcds.2009.23.249

*+*[Abstract](2453)*+*[PDF](1090.9KB)**Abstract:**

Orbital minimization is among the most promising linear scalingalgorithms for electronic structure calculation. However, to achievelinear scaling, one has to truncate the support of the orbitals and thisintroduces many problems, the most important of which is theoccurrence of numerous local minima. In this paper, we introduce a simplemodification of the orbital minimization method, by adding a localizationstep into the algorithm. This localization step selects the most localizedrepresentation of the subspace spanned by the orbitals obtained during theintermediate stages of the iteration process.We show that the addition of the localization step substantially reduces thechances that the iterations get trapped at local minima.

2009, 23(1&2): 265-280
doi: 10.3934/dcds.2009.23.265

*+*[Abstract](2442)*+*[PDF](847.4KB)**Abstract:**

We propose a technique for interactive mesh refinement in regionswhere the solution of a partial differential equation is lessregular. 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.

2009, 23(1&2): 281-298
doi: 10.3934/dcds.2009.23.281

*+*[Abstract](2354)*+*[PDF](246.8KB)**Abstract:**

In this paper, we perform a systematic multiscale analysis forconvection dominated transport equations with a weak diffusion and ahighly oscillatory velocity field. The paper primarily focuses onupscaling linear transport equations. But we also discuss brieflyhow to upscale two-phase miscible flows, in which casethe concentration equation is coupled to the pressure equationin 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 thesmall scale in the heterogeneous media. Due to the presence of thenonlocal memory effect, upscaling a convection dominated transportequation is known to be very difficult. One of the key ideas inderiving a well-posed homogenized equation for the convectiondominated transport equation is to introduce a projection operatorwhich projects the fluctuation onto a suitable subspace. Thisprojection operator corresponds to averaging along the streamlinesof the flow. In the case of linear convection dominated transportequations, we prove the well-posedness of the homogenized equationsand establish rigorous error estimates for our multiscale expansion.

2009, 23(1&2): 299-313
doi: 10.3934/dcds.2009.23.299

*+*[Abstract](2387)*+*[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.

2009, 23(1&2): 315-339
doi: 10.3934/dcds.2009.23.315

*+*[Abstract](2537)*+*[PDF](9194.2KB)**Abstract:**

Our aim in this article is to study the interaction of\textit{boundary layers} and \textit{corner singularities} in thecontext of singularly perturbed convection-diffusion equations. Forthe problems under consideration, we determine a simplified form ofthe corner singularities and show how to use it for the numericalapproximation of such problems in the context of variationalapproximations using the concept of \textit{enriched spaces}.

2009, 23(1&2): 341-365
doi: 10.3934/dcds.2009.23.341

*+*[Abstract](2198)*+*[PDF](296.6KB)**Abstract:**

Assuming minimal regularity assumptions on the data, we revisit the classical problem of findingisometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold.Our approach encompasses metrics having Sobolev regularity and Riemann curvature definedin the distributional sense, only.It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changesfrom point to point.

2009, 23(1&2): 367-380
doi: 10.3934/dcds.2009.23.367

*+*[Abstract](2858)*+*[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.

2009, 23(1&2): 381-397
doi: 10.3934/dcds.2009.23.381

*+*[Abstract](2826)*+*[PDF](218.6KB)**Abstract:**

In thispaper we study the mixed initial-boundary value problem for theequation of time-like extremal surfaces in Minkowski space$R^{1+(1+n)}$ on the strip $R^{+}\times[0,1]$. Under theassumptions that the boundary data are small and decaying, we getthe global existence and uniqueness of classical solutions.

2009, 23(1&2): 399-414
doi: 10.3934/dcds.2009.23.399

*+*[Abstract](2456)*+*[PDF](221.2KB)**Abstract:**

In this paper, we study the exact controllability of a system of twoweakly coupled one-dimensional wave equations with the control actedon only one equation. Using the non harmonic analysis, we establishthe weak observability inequalities, which depend on the ratio ofthe wave propagation speeds. The obtained results are optimal.

2009, 23(1&2): 415-433
doi: 10.3934/dcds.2009.23.415

*+*[Abstract](2491)*+*[PDF](256.4KB)**Abstract:**

This work is concerned with the two-fluidEuler-Maxwell equations for plasmas with small parameters. We study,by means of asymptotic expansions, the zero-relaxation limit, thenon-relativistic limit and the combined non-relativistic and quasi-neutrallimit. For each limit with well-prepared initial data, we show theexistence and uniqueness of an asymptotic expansion up to any order. Forgeneral data, an asymptotic expansion up to order 1 of thenon-relativistic limit is constructed by taking into account the initiallayers. Finally, we discuss the justification of the limits.

2009, 23(1&2): 435-454
doi: 10.3934/dcds.2009.23.435

*+*[Abstract](2535)*+*[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.

2009, 23(1&2): 455-475
doi: 10.3934/dcds.2009.23.455

*+*[Abstract](4470)*+*[PDF](237.6KB)**Abstract:**

In this paper we present results for the existence of classicalsolutions of a hydrodynamical system modeling the flow of nematicliquid crystals. The system consists of a coupled system ofNavier-Stokes equations and various kinematic transport equationsfor the molecular orientations. A formal physical derivation of theinduced elastic stress using least action principle reflects thespecial coupling between the transport and the induced stress terms.The derivation and the analysis of the system falls into a generalenergetic variational framework for complex fluids with elasticeffects due to the presence of nontrivial microstructures.

2009, 23(1&2): 477-494
doi: 10.3934/dcds.2009.23.477

*+*[Abstract](2839)*+*[PDF](251.3KB)**Abstract:**

This paper treats quasilinear elliptic equations indivergence form whose inhomogeneous term is a signed measure. Wefirst prove the existence and continuity of generalized solutions tothe Dirichlet problem. The main result of this paper is a weakconvergence result, extending previous work of the authors forsubharmonic functions and non-negative measures. We also prove auniqueness result for uniformly elliptic operators and for operatorsof $p$-Laplacian type.

2009, 23(1&2): 495-520
doi: 10.3934/dcds.2009.23.495

*+*[Abstract](2267)*+*[PDF](331.5KB)**Abstract:**

The exterior problem arising from the study of a flow past anobstacle is one of the most classical and important subjects in gasdynamics and fluid mechanics. The point of this problem is to assignthe bulk velocity at infinity, which is not a trivial driving forceon the flow so that some non-trivial solution profiles persist. Inthis paper, we consider the exterior problem for the Boltzmannequation when the Mach number of the far field equilibrium state issmall. The result here generalizes the previous one by Ukai-Asano onthe same problem to more general boundary conditions by cruciallyusing the velocity average argument.

2009, 23(1&2): 521-540
doi: 10.3934/dcds.2009.23.521

*+*[Abstract](2404)*+*[PDF](234.4KB)**Abstract:**

We show that stationary statistical properties for uniformlydissipative dynamical systems are upper semi-continuous underregular perturbation and a special type of singular perturbationin time of relaxation type. The results presented are applicableto many physical systems such as the singular limit of infinitePrandtl-Darcy number in the Darcy-Boussinesq system forconvection in porous media, or the large Prandtl asymptotics forthe Boussinesq system.

2009, 23(1&2): 541-560
doi: 10.3934/dcds.2009.23.541

*+*[Abstract](2565)*+*[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 thirdPainlevé transcendents. The connectionformulas are derived by the method of "uniform asymptotics"proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech.Anal., 1998).

2009, 23(1&2): 561-569
doi: 10.3934/dcds.2009.23.561

*+*[Abstract](2803)*+*[PDF](138.1KB)**Abstract:**

Diffusion equations with degenerate nonlinear source terms arisein many different applications, e.g., in the theory of epidemics,in models of cortical spreading depression, and in models ofevaporation and condensation in porous media. In this paper, weconsider 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 ofuniform rest states and show that traveling wave trajectoriesconnecting two stable rest states can exist generically only fordiscrete wave speeds. Furthermore, we show that families oftraveling waves with a continuum of wave speeds cannot exist.

2009, 23(1&2): 571-604
doi: 10.3934/dcds.2009.23.571

*+*[Abstract](2878)*+*[PDF](932.3KB)**Abstract:**

In this paper we study the exact boundary controllability of atrapezoidal time discrete wave equation in a bounded domain. Weprove that the projection of the solution in an appropriate filteredspace is exactly controllable with uniformly bounded cost withrespect to the time-step. In this way, the well-knownexact-controllability property of the wave equation can bereproduced as the limit, as the time step $h\rightarrow 0$, of thecontrollability of projections of the time-discrete one. By dualitythese results are equivalent to deriving uniform observabilityestimates (with respect to $h\rightarrow 0$) within a class ofsolutions of the time-discrete problem in which the high frequencycomponents have been filtered. The later is established by means ofa time-discrete version of the classical multiplier technique. Theoptimality of the order of the filtering parameter is alsoestablished, although a careful analysis of the expected velocity ofpropagation of time-discrete waves indicates that its actual valuecould be improved.

2009, 23(1&2): 605-616
doi: 10.3934/dcds.2009.23.605

*+*[Abstract](1948)*+*[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 thetwo dimensional isentropic Euler system through severalexamples with axially symmetric initial data. We find thatcharacteristics in some cases can be completely absorbed by the sonic curveso that the characteristics vanish tangentially into the sonicboundary, exemplifying a classical scenario of the Keldysh type;however, the characteristics can wraparound the closed sonic curve unboundedly many times, so thatthe domain of determinacy of the hyperbolic characteristic boundaryvalue problem or the Goursat problem exhibit layered annulus structures.As the number of layers increases, the layers become thinner, and thesolution at an interior point of the domain depends eventually on theentire boundary data.

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