May  2018, 23(3): 1347-1361. doi: 10.3934/dcdsb.2018154

Loss of derivatives for hyperbolic boundary problems with constant coefficients

Department of Mathematics, Georgetown University, Washington, DC 20057, USA

Received  January 2017 Revised  October 2017 Published  February 2018

Symmetric hyperbolic systems and constantly hyperbolic systems with constant coefficients and a boundary condition which satisfies a weakened form of the Kreiss-Sakamoto condition are considered. A well-posedness result is established which generalizes a theorem by Chazarain and Piriou for scalar strictly hyperbolic equations and non-characteristic boundaries [3]. The proof is based on an explicit solution of the boundary problem by means of the Fourier-Laplace transform. As long as the operator is symmetric, the boundary is allowed to be characteristic.

Citation: Matthias Eller. Loss of derivatives for hyperbolic boundary problems with constant coefficients. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1347-1361. doi: 10.3934/dcdsb.2018154
References:
[1]

S. Benzoni-GavageF. RoussetD. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1073-1104. doi: 10.1017/S030821050000202X. Google Scholar

[2]

S. L. Campbell, Singular Systems of Differential Equations, volume 40 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 1980. Google Scholar

[3]

J. Chazarain and A. Piriou, Caractérisation des problémes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237. doi: 10.5802/aif.438. Google Scholar

[4]

J. Chazarain and Al. Piriou, Introduction to the Theory of Linear Partial Differential Equations, volume 14 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1982. Translated from the French. Google Scholar

[5]

I. ChueshovI. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Differential Equations, 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. Google Scholar

[6]

J.-F. Coulombel, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 401-443. doi: 10.1016/j.anihpc.2003.04.001. Google Scholar

[7]

J.-F. Coulombel, The hyperbolic region for hyperbolic boundary value problems, Osaka J. Math., 48 (2011), 457-469. Google Scholar

[8]

M. Eller, On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition, Discrete Contin. Dynam. Systems Series S, 2 (2009), 473-481. doi: 10.3934/dcdss.2009.2.473. Google Scholar

[9]

M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions, SIAM Journal of Mathematical Analysis, 4 (2012), 1925-1949. doi: 10.1137/110834652. Google Scholar

[10]

O. GuèsG. MétivierM. Williams and K. Zumbrun, Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems, Comm. Partial Differential Equations, 32 (2007), 579-590. doi: 10.1080/03605300600636804. Google Scholar

[11]

R. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334. Google Scholar

[12]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. Google Scholar

[13]

I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Ⅰ. L2 nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367. doi: 10.1007/BF01765322. Google Scholar

[14]

I. Lasiecka, "Sharp" regularity results for mixed hyperbolic problems of second order, In Differential equations in Banach spaces (Bologna, 1985), volume 1223 of Lecture Notes in Math., pages 160-175. Springer, Berlin, 1986. Google Scholar

[15]

A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504. Google Scholar

[16]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702. doi: 10.1112/S0024609300007517. Google Scholar

[17]

G. Métivier, On the L2 well-posedness of hyperbolic boundary value problems, Preprint, 2014.Google Scholar

[18]

G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations, 211 (2005), 61-134. doi: 10.1016/j.jde.2004.06.002. Google Scholar

[19]

T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123. doi: 10.14492/hokmj/1381758116. Google Scholar

[20]

J. V. Ralston, Note on a paper by Kreiss, Comm. Pure Appl. Math., 24 (1971), 759-762. doi: 10.1002/cpa.3160240603. Google Scholar

[21]

R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge, 1982. Translated from the Japanese by Katsumi Miyahara. Google Scholar

[22]

D. Serre, Systems of Conservation Laws. 2, Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by Ⅰ. N. Sneddon. Google Scholar

[23]

G. W. Stewart and J. Guang Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, 1990. Google Scholar

show all references

References:
[1]

S. Benzoni-GavageF. RoussetD. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1073-1104. doi: 10.1017/S030821050000202X. Google Scholar

[2]

S. L. Campbell, Singular Systems of Differential Equations, volume 40 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass., 1980. Google Scholar

[3]

J. Chazarain and A. Piriou, Caractérisation des problémes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237. doi: 10.5802/aif.438. Google Scholar

[4]

J. Chazarain and Al. Piriou, Introduction to the Theory of Linear Partial Differential Equations, volume 14 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1982. Translated from the French. Google Scholar

[5]

I. ChueshovI. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Differential Equations, 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. Google Scholar

[6]

J.-F. Coulombel, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 401-443. doi: 10.1016/j.anihpc.2003.04.001. Google Scholar

[7]

J.-F. Coulombel, The hyperbolic region for hyperbolic boundary value problems, Osaka J. Math., 48 (2011), 457-469. Google Scholar

[8]

M. Eller, On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition, Discrete Contin. Dynam. Systems Series S, 2 (2009), 473-481. doi: 10.3934/dcdss.2009.2.473. Google Scholar

[9]

M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions, SIAM Journal of Mathematical Analysis, 4 (2012), 1925-1949. doi: 10.1137/110834652. Google Scholar

[10]

O. GuèsG. MétivierM. Williams and K. Zumbrun, Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems, Comm. Partial Differential Equations, 32 (2007), 579-590. doi: 10.1080/03605300600636804. Google Scholar

[11]

R. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334. Google Scholar

[12]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. Google Scholar

[13]

I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Ⅰ. L2 nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367. doi: 10.1007/BF01765322. Google Scholar

[14]

I. Lasiecka, "Sharp" regularity results for mixed hyperbolic problems of second order, In Differential equations in Banach spaces (Bologna, 1985), volume 1223 of Lecture Notes in Math., pages 160-175. Springer, Berlin, 1986. Google Scholar

[15]

A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504. Google Scholar

[16]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702. doi: 10.1112/S0024609300007517. Google Scholar

[17]

G. Métivier, On the L2 well-posedness of hyperbolic boundary value problems, Preprint, 2014.Google Scholar

[18]

G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations, 211 (2005), 61-134. doi: 10.1016/j.jde.2004.06.002. Google Scholar

[19]

T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123. doi: 10.14492/hokmj/1381758116. Google Scholar

[20]

J. V. Ralston, Note on a paper by Kreiss, Comm. Pure Appl. Math., 24 (1971), 759-762. doi: 10.1002/cpa.3160240603. Google Scholar

[21]

R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge, 1982. Translated from the Japanese by Katsumi Miyahara. Google Scholar

[22]

D. Serre, Systems of Conservation Laws. 2, Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by Ⅰ. N. Sneddon. Google Scholar

[23]

G. W. Stewart and J. Guang Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, 1990. Google Scholar

[1]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[2]

Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403

[3]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547

[4]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271

[5]

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

[6]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205

[7]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814

[8]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590

[9]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[10]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107

[11]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865

[12]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[13]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431

[14]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257

[15]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51

[16]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[17]

Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305

[18]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31

[19]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595

[20]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (53)
  • HTML views (272)
  • Cited by (0)

Other articles
by authors

[Back to Top]