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August  2014, 19(6): 1627-1665. doi: 10.3934/dcdsb.2014.19.1627

## The linear hyperbolic initial and boundary value problems in a domain with corners

 1 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405 2 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405

Received  October 2013 Revised  March 2014 Published  June 2014

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.
Citation: Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627
##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007. [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics, Physics of Plasmas, 9 (2002), 1130-1136. doi: 10.1063/1.1463415. [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194, Springer-Verlag, 2000. [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc., 55 (1944), 132-151. doi: 10.1090/S0002-9947-1944-0009701-0. [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline, Astrophys. J. Lett., 544 (2000), L79-L82. doi: 10.1086/317291. [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985. doi: 10.1137/1.9781611972030. [8] L. Hörmander, Weak and Strong Extensions of Differential Operators, Comm. Pure Appl. Math., 14 (1961), 371-379. [9] R. A. Horn and C. R. Johnson, Matrix Analysis, Second ed., Cambridge University Press, Cambridge, 2013. [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1974 (Third printing of the revised edition of 1957, AMS Colloquium Publications, vol. XXXI). [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: Boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063. doi: 10.1007/s00205-013-0702-0. [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle, Matrix Editions, Ithaca, NY, 2006. [13] I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304. [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner, Funkcialaj Ekvacioj, 23 (1980), 171-195. [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press Oxford University Press, New York, 1998, Compressible models, Oxford Science Publications. [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York, 1972. [18] J. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594. [19] J. Li and J. Zha, Linear Algebra, Univ. of Sci. & Tech. of China Press, P.R. China, 1988, in Chinese. [20] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. doi: 10.1137/0135035. [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5. [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175. doi: 10.1090/S0002-9947-1974-0352715-0. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001. [25] W. Rudin, Functional Analysis, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., 26 (1977), 1-39. doi: 10.1512/iumj.1977.26.26001. [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives, Math. Sb., 17 (1945), 193-210, in Russia. [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations, Physics of Plasmas, 8 (2001), 3293-3304. doi: 10.1063/1.1379045. [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. doi: 10.1115/1.3424338. [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2. [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices, Linear Algebra and its Applications, 7 (1973), 281-289. doi: 10.1016/S0024-3795(73)80001-1. [33] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London, 1962. [34] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2. [35] K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, Berlin, 1995.

show all references

##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007. [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics, Physics of Plasmas, 9 (2002), 1130-1136. doi: 10.1063/1.1463415. [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194, Springer-Verlag, 2000. [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc., 55 (1944), 132-151. doi: 10.1090/S0002-9947-1944-0009701-0. [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline, Astrophys. J. Lett., 544 (2000), L79-L82. doi: 10.1086/317291. [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985. doi: 10.1137/1.9781611972030. [8] L. Hörmander, Weak and Strong Extensions of Differential Operators, Comm. Pure Appl. Math., 14 (1961), 371-379. [9] R. A. Horn and C. R. Johnson, Matrix Analysis, Second ed., Cambridge University Press, Cambridge, 2013. [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1974 (Third printing of the revised edition of 1957, AMS Colloquium Publications, vol. XXXI). [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: Boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063. doi: 10.1007/s00205-013-0702-0. [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle, Matrix Editions, Ithaca, NY, 2006. [13] I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304. [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner, Funkcialaj Ekvacioj, 23 (1980), 171-195. [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press Oxford University Press, New York, 1998, Compressible models, Oxford Science Publications. [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York, 1972. [18] J. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594. [19] J. Li and J. Zha, Linear Algebra, Univ. of Sci. & Tech. of China Press, P.R. China, 1988, in Chinese. [20] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. doi: 10.1137/0135035. [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5. [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175. doi: 10.1090/S0002-9947-1974-0352715-0. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001. [25] W. Rudin, Functional Analysis, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., 26 (1977), 1-39. doi: 10.1512/iumj.1977.26.26001. [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives, Math. Sb., 17 (1945), 193-210, in Russia. [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations, Physics of Plasmas, 8 (2001), 3293-3304. doi: 10.1063/1.1379045. [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. doi: 10.1115/1.3424338. [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2. [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices, Linear Algebra and its Applications, 7 (1973), 281-289. doi: 10.1016/S0024-3795(73)80001-1. [33] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London, 1962. [34] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2. [35] K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, Berlin, 1995.
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