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Artigos de revistas sobre o tema "Weakly hyperbolic systems"

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Autor: Grafiati

Publicado: 25 de maio de 2024

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1

Arbieto, Alexander, André Junqueira e Bruno Santiago. "On Weakly Hyperbolic Iterated Function Systems". Bulletin of the Brazilian Mathematical Society, New Series 48, n.º 1 (4 de outubro de 2016): 111–40. http://dx.doi.org/10.1007/s00574-016-0018-4.

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2

YONEDA, GEN, e HISA-AKI SHINKAI. "CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY". International Journal of Modern Physics D 09, n.º 01 (fevereiro de 2000): 13–34. http://dx.doi.org/10.1142/s0218271800000037.

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Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

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3

Krylovas, A., e R. Čiegis. "Asymptotic Approximation of Hyperbolic Weakly Nonlinear Systems". Journal of Nonlinear Mathematical Physics 8, n.º 4 (janeiro de 2001): 458–70. http://dx.doi.org/10.2991/jnmp.2001.8.4.2.

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4

Spagnolo, Sergio, e Giovanni Taglialatela. "Analytic Propagation for Nonlinear Weakly Hyperbolic Systems". Communications in Partial Differential Equations 35, n.º 12 (4 de novembro de 2010): 2123–63. http://dx.doi.org/10.1080/03605300903440490.

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5

Colombini, F., e Guy Métivier. "The Cauchy problem for weakly hyperbolic systems". Communications in Partial Differential Equations 43, n.º 1 (8 de dezembro de 2017): 25–46. http://dx.doi.org/10.1080/03605302.2017.1399906.

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6

Arbieto, Alexander, Carlos Matheus e Maria José Pacifico. "The Bernoulli Property for Weakly Hyperbolic Systems". Journal of Statistical Physics 117, n.º 1/2 (outubro de 2004): 243–60. http://dx.doi.org/10.1023/b:joss.0000044058.99450.c9.

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7

D'Ancona, Piero, Tamotu Kinoshita e Sergio Spagnolo. "Weakly hyperbolic systems with Hölder continuous coefficients". Journal of Differential Equations 203, n.º 1 (agosto de 2004): 64–81. http://dx.doi.org/10.1016/j.jde.2004.03.016.

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8

Souza, Rafael R. "Sub-actions for weakly hyperbolic one-dimensional systems". Dynamical Systems 18, n.º 2 (junho de 2003): 165–79. http://dx.doi.org/10.1080/1468936031000136126.

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9

Alabau-Boussouira, Fatiha. "Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems". SIAM Journal on Control and Optimization 41, n.º 2 (janeiro de 2002): 511–41. http://dx.doi.org/10.1137/s0363012901385368.

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10

DREHER, MICHAEL, e INGO WITT. "ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER". Communications in Contemporary Mathematics 07, n.º 06 (dezembro de 2005): 809–37. http://dx.doi.org/10.1142/s0219199705001969.

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For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.

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11

Jiang, Kai. "Local normal forms of smooth weakly hyperbolic integrable systems". Regular and Chaotic Dynamics 21, n.º 1 (janeiro de 2016): 18–23. http://dx.doi.org/10.1134/s1560354716010020.

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12

Melo, Ítalo. "On $$\mathbb {P}$$ P -Weakly Hyperbolic Iterated Function Systems". Bulletin of the Brazilian Mathematical Society, New Series 48, n.º 4 (30 de maio de 2017): 717–32. http://dx.doi.org/10.1007/s00574-017-0042-z.

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13

Bessa, Mário, Manseob Lee e Sandra Vaz. "Stable weakly shadowable volume-preserving systems are volume-hyperbolic". Acta Mathematica Sinica, English Series 30, n.º 6 (15 de maio de 2014): 1007–20. http://dx.doi.org/10.1007/s10114-014-3093-8.

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14

DAFERMOS, C. M. "HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH WEAK DISSIPATION". Journal of Hyperbolic Differential Equations 03, n.º 03 (setembro de 2006): 505–27. http://dx.doi.org/10.1142/s0219891606000884.

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Global BV solutions are constructed to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a rich family of entropies and source that is merely weakly dissipative, of the type induced by relaxation mechanisms.

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15

SHAO, ZHI-QIANG. "GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS". Mathematical Models and Methods in Applied Sciences 19, n.º 07 (julho de 2009): 1099–138. http://dx.doi.org/10.1142/s0218202509003735.

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In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

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16

Chen, Gui-Qiang, Wei Xiang e Yongqian Zhang. "Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws". Communications in Partial Differential Equations 38, n.º 11 (2 de novembro de 2013): 1936–70. http://dx.doi.org/10.1080/03605302.2013.828229.

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17

Krylov, A. V. "Averaging of weakly nonlinear hyperbolic systems with nonuniform integral means". Ukrainian Mathematical Journal 43, n.º 5 (maio de 1991): 566–73. http://dx.doi.org/10.1007/bf01058542.

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18

Rasskazov, I. O. "The Riemann Problem for Weakly Perturbed 2 × 2 Hyperbolic Systems". Journal of Mathematical Sciences 122, n.º 5 (agosto de 2004): 3564–71. http://dx.doi.org/10.1023/b:joth.0000034036.97955.a8.

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19

Jiang, Ning, e C. David Levermore. "Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy". Archive for Rational Mechanics and Analysis 201, n.º 2 (28 de maio de 2011): 377–412. http://dx.doi.org/10.1007/s00205-010-0361-3.

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20

Bondarev, B. V. "Averaging in hyperbolic systems subject to weakly dependent random perturbations". Ukrainian Mathematical Journal 44, n.º 8 (agosto de 1992): 915–23. http://dx.doi.org/10.1007/bf01057110.

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21

REULA, OSCAR A. "STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY". Journal of Hyperbolic Differential Equations 01, n.º 02 (junho de 2004): 251–69. http://dx.doi.org/10.1142/s0219891604000111.

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We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.

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22

Krylovas, A., e R. Čiegis. "A REVIEW OF NUMERICAL ASYMPTOTIC AVERAGING FOR WEAKLY NONLINEAR HYPERBOLIC WAVES". Mathematical Modelling and Analysis 9, n.º 3 (30 de setembro de 2004): 209–22. http://dx.doi.org/10.3846/13926292.2004.9637254.

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We present an overview of averaging method for solving weakly nonlinear hyperbolic systems. An asymptotic solution is constructed, which is uniformly valid in the “large” domain of variables t + |x| ∼ O(ϵ –1). Using this method we obtain the averaged system, which disintegrates into independent equations for the nonresonant systems. A scheme for theoretical justification of such algorithms is given and examples are presented. The averaged systems with periodic solutions are investigated for the following problems of mathematical physics: shallow water waves, gas dynamics and elastic waves. In the resonant case the averaged systems must be solved numerically. They are approximated by the finite difference schemes and the results of numerical experiments are presented.

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23

Begun, N. A. "Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems". Vestnik St. Petersburg University: Mathematics 48, n.º 1 (janeiro de 2015): 1–8. http://dx.doi.org/10.3103/s1063454115010033.

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24

Li, Ta-Tsien, e Yue-Jun Peng. "Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form". Nonlinear Analysis: Theory, Methods & Applications 55, n.º 7-8 (dezembro de 2003): 937–49. http://dx.doi.org/10.1016/j.na.2003.08.010.

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25

Pesin, Ya B., e Ya G. Sinai. "Space-time chaos in the system of weakly interacting hyperbolic systems". Journal of Geometry and Physics 5, n.º 3 (1988): 483–92. http://dx.doi.org/10.1016/0393-0440(88)90035-6.

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26

Rohde, Ch. "Entropy solutions for weakly coupled hyperbolic systems in several space dimensions". Zeitschrift für angewandte Mathematik und Physik 49, n.º 3 (1998): 470. http://dx.doi.org/10.1007/s000000050102.

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27

GOSSE, LAURENT, SHI JIN e XIANTAO LI. "TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION". Mathematical Models and Methods in Applied Sciences 13, n.º 12 (dezembro de 2003): 1689–723. http://dx.doi.org/10.1142/s0218202503003082.

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Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schrödinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

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28

Krylovas, Aleksandras. "Application of the method of stationary phase to weakly nonlinear hyperbolic systems asymptotic solving". Lietuvos matematikos rinkinys 44 (17 de dezembro de 2004): 164–68. http://dx.doi.org/10.15388/lmr.2004.31907.

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29

Krylovas, A. "ASYMPTOTIC METHOD FOR APPROXIMATION OF RESONANT INTERACTION OF NONLINEAR MULTIDIMENSIONAL HYPERBOLIC WAVES". Mathematical Modelling and Analysis 13, n.º 1 (31 de março de 2008): 47–54. http://dx.doi.org/10.3846/1392-6292.2008.13.47-54.

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A method of averaging along characteristics of weakly nonlinear hyperbolic systems, which was presented in earlier works of the author for one dimensional waves, is generalized for some cases of multidimensional wave problems. In this work we consider such systems and discuss a way to use the internal averaging along characteristics for new problems of asymptotical integration.

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30

Alves, José, Carla Dias, Stefano Luzzatto e Vilton Pinheiro. "SRB measures for partially hyperbolic systems whose central direction is weakly expanding". Journal of the European Mathematical Society 19, n.º 10 (2017): 2911–46. http://dx.doi.org/10.4171/jems/731.

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31

Demengel, F., e J. Rauch. "Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable". Proceedings of the Edinburgh Mathematical Society 33, n.º 3 (outubro de 1990): 443–60. http://dx.doi.org/10.1017/s0013091500004855.

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We study systems which in characteristic coordinates have the formwhere A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.Simple weak converge of the initial data does not imply weak convergence of the solutions.

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32

Gao, Jing, e Yao-Lin Jiang. "A periodic wavelet method for the second kind of the logarithmic integral equation". Bulletin of the Australian Mathematical Society 76, n.º 3 (dezembro de 2007): 321–36. http://dx.doi.org/10.1017/s0004972700039721.

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A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

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33

Kinoshita, Tamotu. "On the Cauchy Problem with small analytic data for nonlinear weakly hyperbolic systems". Tsukuba Journal of Mathematics 21, n.º 2 (outubro de 1997): 397–420. http://dx.doi.org/10.21099/tkbjm/1496163249.

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34

Fitzgibbon, W. E., e Michel Langlais. "Weakly coupled hyperbolic systems modeling the circulation of FeLV in structured feline populations". Mathematical Biosciences 165, n.º 1 (maio de 2000): 79–95. http://dx.doi.org/10.1016/s0025-5564(00)00011-0.

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35

Garg, Naveen Kumar. "A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems". Numerical Algorithms 83, n.º 3 (15 de maio de 2019): 1091–121. http://dx.doi.org/10.1007/s11075-019-00717-7.

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36

Barkwell, Lawrence, Peter Lancaster e Alexander S. Markus. "Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum". Canadian Journal of Mathematics 44, n.º 1 (1 de dezembro de 1991): 42–53. http://dx.doi.org/10.4153/cjm-1992-002-2.

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AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

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37

Williams, Mark. "Weakly stable hyperbolic boundary problems with large oscillatory coefficients: Simple cascades". Journal of Hyperbolic Differential Equations 17, n.º 01 (março de 2020): 141–83. http://dx.doi.org/10.1142/s0219891620500058.

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We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, [Formula: see text], compared to the wavelength of the oscillations, [Formula: see text]. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of [Formula: see text]. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.

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38

Benzoni-Gavage, Sylvie, Frédéric Rousset, Denis Serre e K. Zumbrun. "Generic types and transitions in hyperbolic initial–boundary-value problems". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, n.º 5 (outubro de 2002): 1073–104. http://dx.doi.org/10.1017/s030821050000202x.

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The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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39

Benzoni-Gavage, Sylvie, Frédéric Rousset, Denis Serre e K. Zumbrun. "Generic types and transitions in hyperbolic initial–boundary-value problems". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, n.º 5 (outubro de 2002): 1073–104. http://dx.doi.org/10.1017/s0308210502000537.

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The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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40

Morisse, Baptiste. "On hyperbolicity and Gevrey well-posedness. Part three: a model of weakly hyperbolic systems". Indiana University Mathematics Journal 70, n.º 2 (2021): 743–80. http://dx.doi.org/10.1512/iumj.2021.70.8198.

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41

Corli, Andrea. "Weakly non-linear geometric optics for hyperbolic systems of conservation laws with shock waves". Asymptotic Analysis 10, n.º 2 (1995): 117–72. http://dx.doi.org/10.3233/asy-1995-10202.

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42

Rohde, Christian. "Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D". Numerische Mathematik 81, n.º 1 (1 de novembro de 1998): 85–123. http://dx.doi.org/10.1007/s002110050385.

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43

Margenstern, Maurice. "A Weakly Universal Cellular Automaton in the Heptagrid of the Hyperbolic Plane". Complex Systems 27, n.º 4 (15 de dezembro de 2018): 315–54. http://dx.doi.org/10.25088/complexsystems.27.4.315.

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44

Korsch, Andrea, e Dietmar Kröner. "On existence and uniqueness of entropy solutions of weakly coupled hyperbolic systems on evolving surfaces". Computers & Fluids 169 (junho de 2018): 296–308. http://dx.doi.org/10.1016/j.compfluid.2017.08.021.

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45

Qu, Peng, e Cunming Liu. "Global classical solutions to partially dissipative quasilinear hyperbolic systems with one weakly linearly degenerate characteristic". Chinese Annals of Mathematics, Series B 33, n.º 3 (maio de 2012): 333–50. http://dx.doi.org/10.1007/s11401-012-0715-2.

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46

Benoit, Antoine. "WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives". Journal of Hyperbolic Differential Equations 18, n.º 03 (setembro de 2021): 557–608. http://dx.doi.org/10.1142/s0219891621500181.

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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.

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47

Alabau-Boussouira, Fatiha. "A Two-Level Energy Method for Indirect Boundary Observability and Controllability of Weakly Coupled Hyperbolic Systems". SIAM Journal on Control and Optimization 42, n.º 3 (janeiro de 2003): 871–906. http://dx.doi.org/10.1137/s0363012902402608.

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48

Gourdin, Daniel, e Todor Gramchev. "Global Cauchy problems on Rn for weakly hyperbolic systems with coefficients admitting superlinear growth for |x| → ∞". Bulletin des Sciences Mathématiques 150 (fevereiro de 2019): 35–61. http://dx.doi.org/10.1016/j.bulsci.2016.01.002.

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Dai, Wenrong. "Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy*". Chinese Annals of Mathematics, Series B 27, n.º 3 (junho de 2006): 263–86. http://dx.doi.org/10.1007/s11401-004-0523-4.

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