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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 & 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).   Google Scholar [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics,, Physics of Plasmas, 9 (2002), 1130.  doi: 10.1063/1.1463415.  Google Scholar [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math., (2000).   Google Scholar [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator,, Trans. Amer. Math. Soc., 55 (1944), 132.  doi: 10.1090/S0002-9947-1944-0009701-0.  Google Scholar [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline,, Astrophys. J. Lett., 544 (2000).  doi: 10.1086/317291.  Google Scholar [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-0713-9.  Google Scholar [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).  doi: 10.1137/1.9781611972030.  Google Scholar [8] L. Hörmander, Weak and Strong Extensions of Differential Operators,, Comm. Pure Appl. Math., 14 (1961), 371.   Google Scholar [9] R. A. Horn and C. R. Johnson, Matrix Analysis,, Second ed., (2013).   Google Scholar [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1974).   Google Scholar [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.  doi: 10.1007/s00205-013-0702-0.  Google Scholar [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,, Vol. 1. Teichmüller theory, (2006).   Google Scholar [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.  doi: 10.1002/cpa.3160240304.  Google Scholar [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner,, Funkcialaj Ekvacioj, 23 (1980), 171.   Google Scholar [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I,, Springer-Verlag, (1972).   Google Scholar [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.   Google Scholar [19] J. Li and J. Zha, Linear Algebra,, Univ. of Sci. & Tech. of China Press, (1988).   Google Scholar [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.  doi: 10.1137/0135035.  Google Scholar [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I,, Trans. Amer. Math. Soc., 176 (1973), 141.  doi: 10.1090/S0002-9947-1973-0320539-5.  Google Scholar [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II,, Trans. Amer. Math. Soc., 198 (1974), 155.  doi: 10.1090/S0002-9947-1974-0352715-0.  Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [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.  doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [25] W. Rudin, Functional Analysis,, Second ed., (1991).   Google Scholar [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges,, Indiana Univ. Math. J., 26 (1977), 1.  doi: 10.1512/iumj.1977.26.26001.  Google Scholar [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives,, Math. Sb., 17 (1945), 193.   Google Scholar [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations,, Physics of Plasmas, 8 (2001), 3293.  doi: 10.1063/1.1379045.  Google Scholar [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner,, Funkcialaj Ekvacioj, 21 (1978), 249.   Google Scholar [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001).  doi: 10.1115/1.3424338.  Google Scholar [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations,, J. Atmospheric Sci., 60 (2003), 2647.  doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.  Google Scholar [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices,, Linear Algebra and its Applications, 7 (1973), 281.  doi: 10.1016/S0024-3795(73)80001-1.  Google Scholar [33] I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (1962).   Google Scholar [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.  doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.  Google Scholar [35] K. Yosida, Functional Analysis,, Sixth ed., (1995).   Google Scholar

show all references

##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations,, Oxford University Press, (2007).   Google Scholar [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics,, Physics of Plasmas, 9 (2002), 1130.  doi: 10.1063/1.1463415.  Google Scholar [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math., (2000).   Google Scholar [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator,, Trans. Amer. Math. Soc., 55 (1944), 132.  doi: 10.1090/S0002-9947-1944-0009701-0.  Google Scholar [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline,, Astrophys. J. Lett., 544 (2000).  doi: 10.1086/317291.  Google Scholar [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-0713-9.  Google Scholar [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).  doi: 10.1137/1.9781611972030.  Google Scholar [8] L. Hörmander, Weak and Strong Extensions of Differential Operators,, Comm. Pure Appl. Math., 14 (1961), 371.   Google Scholar [9] R. A. Horn and C. R. Johnson, Matrix Analysis,, Second ed., (2013).   Google Scholar [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1974).   Google Scholar [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.  doi: 10.1007/s00205-013-0702-0.  Google Scholar [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,, Vol. 1. Teichmüller theory, (2006).   Google Scholar [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.  doi: 10.1002/cpa.3160240304.  Google Scholar [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277.  doi: 10.1002/cpa.3160230304.  Google Scholar [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner,, Funkcialaj Ekvacioj, 23 (1980), 171.   Google Scholar [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I,, Springer-Verlag, (1972).   Google Scholar [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.   Google Scholar [19] J. Li and J. Zha, Linear Algebra,, Univ. of Sci. & Tech. of China Press, (1988).   Google Scholar [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.  doi: 10.1137/0135035.  Google Scholar [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I,, Trans. Amer. Math. Soc., 176 (1973), 141.  doi: 10.1090/S0002-9947-1973-0320539-5.  Google Scholar [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II,, Trans. Amer. Math. Soc., 198 (1974), 155.  doi: 10.1090/S0002-9947-1974-0352715-0.  Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [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.  doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [25] W. Rudin, Functional Analysis,, Second ed., (1991).   Google Scholar [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges,, Indiana Univ. Math. J., 26 (1977), 1.  doi: 10.1512/iumj.1977.26.26001.  Google Scholar [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives,, Math. Sb., 17 (1945), 193.   Google Scholar [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations,, Physics of Plasmas, 8 (2001), 3293.  doi: 10.1063/1.1379045.  Google Scholar [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner,, Funkcialaj Ekvacioj, 21 (1978), 249.   Google Scholar [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001).  doi: 10.1115/1.3424338.  Google Scholar [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations,, J. Atmospheric Sci., 60 (2003), 2647.  doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.  Google Scholar [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices,, Linear Algebra and its Applications, 7 (1973), 281.  doi: 10.1016/S0024-3795(73)80001-1.  Google Scholar [33] I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (1962).   Google Scholar [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.  doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.  Google Scholar [35] K. Yosida, Functional Analysis,, Sixth ed., (1995).   Google Scholar
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