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  January 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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

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