• Previous Article
    Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors
  • DCDS-B Home
  • This Issue
  • Next Article
    Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations
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).

[2]

P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics,, Physics of Plasmas, 9 (2002), 1130. doi: 10.1063/1.1463415.

[3]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math., (2000).

[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.

[5]

P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline,, Astrophys. J. Lett., 544 (2000). doi: 10.1086/317291.

[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.

[7]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). doi: 10.1137/1.9781611972030.

[8]

L. Hörmander, Weak and Strong Extensions of Differential Operators,, Comm. Pure Appl. Math., 14 (1961), 371.

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Second ed., (2013).

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1974).

[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.

[12]

J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,, Vol. 1. Teichmüller theory, (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. doi: 10.1002/cpa.3160240304.

[14]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277. 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.

[16]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2,, Oxford Lecture Series in Mathematics and its Applications, (1998).

[17]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I,, Springer-Verlag, (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.

[19]

J. Li and J. Zha, Linear Algebra,, Univ. of Sci. & Tech. of China Press, (1988).

[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.

[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.

[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.

[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. doi: 10.1016/j.matpur.2007.12.001.

[25]

W. Rudin, Functional Analysis,, Second ed., (1991).

[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.

[27]

B. V. Shabat, On a generalized solution to a system of equations in partial derivatives,, Math. Sb., 17 (1945), 193.

[28]

H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations,, Physics of Plasmas, 8 (2001), 3293. doi: 10.1063/1.1379045.

[29]

M. Taniguchi, Mixed problem for wave equation in the domain with a corner,, Funkcialaj Ekvacioj, 21 (1978), 249.

[30]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001). 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. 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. doi: 10.1016/S0024-3795(73)80001-1.

[33]

I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (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. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.

[35]

K. Yosida, Functional Analysis,, Sixth ed., (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. doi: 10.1063/1.1463415.

[3]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math., (2000).

[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.

[5]

P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline,, Astrophys. J. Lett., 544 (2000). doi: 10.1086/317291.

[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.

[7]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). doi: 10.1137/1.9781611972030.

[8]

L. Hörmander, Weak and Strong Extensions of Differential Operators,, Comm. Pure Appl. Math., 14 (1961), 371.

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Second ed., (2013).

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1974).

[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.

[12]

J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics,, Vol. 1. Teichmüller theory, (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. doi: 10.1002/cpa.3160240304.

[14]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277. 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.

[16]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2,, Oxford Lecture Series in Mathematics and its Applications, (1998).

[17]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I,, Springer-Verlag, (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.

[19]

J. Li and J. Zha, Linear Algebra,, Univ. of Sci. & Tech. of China Press, (1988).

[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.

[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.

[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.

[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. doi: 10.1016/j.matpur.2007.12.001.

[25]

W. Rudin, Functional Analysis,, Second ed., (1991).

[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.

[27]

B. V. Shabat, On a generalized solution to a system of equations in partial derivatives,, Math. Sb., 17 (1945), 193.

[28]

H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations,, Physics of Plasmas, 8 (2001), 3293. doi: 10.1063/1.1379045.

[29]

M. Taniguchi, Mixed problem for wave equation in the domain with a corner,, Funkcialaj Ekvacioj, 21 (1978), 249.

[30]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001). 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. 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. doi: 10.1016/S0024-3795(73)80001-1.

[33]

I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (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. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.

[35]

K. Yosida, Functional Analysis,, Sixth ed., (1995).

[1]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59

[2]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

[3]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024

[4]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967

[5]

Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065

[6]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135

[7]

Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63

[8]

Jitao Liu. On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1179-1191. doi: 10.3934/cpaa.2016.15.1179

[9]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212

[10]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

[11]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319

[12]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917

[13]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161

[14]

Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327

[15]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[16]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[17]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[18]

Davide Guidetti. Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5107-5141. doi: 10.3934/dcds.2013.33.5107

[19]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917

[20]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]