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Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
1. Introduction....................................................................................................................................................5
2. Notations and preliminaries .........................................................................................................................11
2.1. Function spaces and spaces of distributions............................................................................................11
2.2. Perturbations of linear operators.............................................................................................................20
2.3. Properties of matrices..............................................................................................................................24
3. A boundary value problem for a linear hyperbolic system of the first order in a halfspace...........................26
3.1. Assumptions.............................................................................................................................................27
3.2. Construction of a symmetrizer..................................................................................................................31
3.3. An estimate of a solution of the boundary value problem (3.1)-(3.2) in $L²_{η}-norm$............................57
3.4. An estimate of a solution of the problem formally adjoint to (3.1)-(3.2) in $L²_{-η}-norm$.......................85
3.5. Existence and uniqueness of solution of the boundary value problem (3.1)-(3.2)....................................91
3.6. An estimate of a solution of the Cauchy problem for system (3.1) in $H^{s}_{η}$-norm...........................95
4. A mixed problem for a system of linear hyperbolic equations of the first order in a halfspace....................101
4.1. A mixed problem with nonzero initial condition........................................................................................101
4.2. A mixed problem with zero initial condition..............................................................................................127
5. A mixed problem for a system of linear hyperbolic equations of the first order in a bounded domain.........128
6. A mixed problem for a nonlinear system of hyperbolic equations of the first order.....................................141
References....................................................................................................................................................145
1. Introduction....................................................................................................................................................5
2. Notations and preliminaries .........................................................................................................................11
2.1. Function spaces and spaces of distributions............................................................................................11
2.2. Perturbations of linear operators.............................................................................................................20
2.3. Properties of matrices..............................................................................................................................24
3. A boundary value problem for a linear hyperbolic system of the first order in a halfspace...........................26
3.1. Assumptions.............................................................................................................................................27
3.2. Construction of a symmetrizer..................................................................................................................31
3.3. An estimate of a solution of the boundary value problem (3.1)-(3.2) in $L²_{η}-norm$............................57
3.4. An estimate of a solution of the problem formally adjoint to (3.1)-(3.2) in $L²_{-η}-norm$.......................85
3.5. Existence and uniqueness of solution of the boundary value problem (3.1)-(3.2)....................................91
3.6. An estimate of a solution of the Cauchy problem for system (3.1) in $H^{s}_{η}$-norm...........................95
4. A mixed problem for a system of linear hyperbolic equations of the first order in a halfspace....................101
4.1. A mixed problem with nonzero initial condition........................................................................................101
4.2. A mixed problem with zero initial condition..............................................................................................127
5. A mixed problem for a system of linear hyperbolic equations of the first order in a bounded domain.........128
6. A mixed problem for a nonlinear system of hyperbolic equations of the first order.....................................141
References....................................................................................................................................................145
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
311
Liczba stron
146
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXI
Daty
wydano
1991
otrzymano
1990-10-11
Twórcy
autor
- Institute of Mathematics, Technical University of Łódź, Al. Politechniki 11, 93-590 Łódź, Poland
Bibliografia
- [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
- [2] M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1984), 159-184.
- [3] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representation of Functions and Imbedding Theorems, Nauka, Moscow 1975 (in Russian).
- [4] Yu. V. Egorov, Linear Differential Equations of Principal Type, Nauka, Moscow 1984 (in Russian).
- [5] K. O. Friedrichs and P. D. Lax, Boundary value problems for first order operators, Comm. Pure Appl. Math. 18 (1965), 355-388.
- [6] F. R. Gantmacher, Matrix Theory, Nauka, Moscow 1954 (in Russian).
- [7] L. Gårding, Solution directe du problème de Cauchy pour les équations hyperboliques, in: Colloq. Internat. CNRS 71, Nancy 1956, 71-90.
- [8] R. Hersh, Mixed problems in several variables, J. Math. Mech. 12 (1963), 317-334.
- [9] L. Hörmander, Linear Partial Differential Operators, Springer, 1963.
- [10] T. Kato, Perturbation Theory for Linear Operators, Springer, 1976.
- [11] T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181-205.
- [12] H.-O. Kreiss, Über sachgemässe Cauchy probleme, Math. Scand. 13 (1963), 109-128.
- [13] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298.
- [14] P. D. Lax and L. Nirenberg, On stability for difference schemes, a sharp form of Gårding's inequality, ibid. 19 (1966), 473-492.
- [15] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, ibid. 13 (1960), 427-455.
- [16] Li Dening, Nonlinear initial-boundary value problem for quasilinear hyperbolic system, Chinese Ann. Math. 7B (2) (1986), 147-159.
- [17] J. L. Lions et, E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1968.
- [18] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, 1984.
- [19] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), 607-675.
- [20] M. Nagumo, Lectures on the Modern Theory of Partial Differential Equations, Mir, Moscow 1967 (in Russian).
- [21] J. Rauch, L² is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math. 25 (1972), 265-285.
- [22] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303-318.
- [23] R. Sakamoto, Mixed problems for hyperbolic equations, I, II, J. Math. Kyoto Univ. 10 (1970), 349-373, 403-417.
- [24] D. S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1972), 1113-1129.
- [25] M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, 1981.
- [26] E. Zadrzyńska, Noncharacteristic mixed problem for nonlinear hyperbolic systems of the first order with zero initial condition, Bull. Polish Acad. Sci. Tech., to appear.
- [27] E. Zadrzyńska, Noncharacteristic mixed problem for nonlinear strictly hyperbolic systems of the first order with nonzero initial condition, ibid., to appear.
- [28] W. M. Zajączkowski, Noncharacteristic mixed problems for nonlinear symmetric hyperbolic equations, Math. Methods Appl. Sci. 11 (1989), 139-168.
- [29] W. M. Zajączkowski, Initial-boundary value problems for nonlinear symmetric systems of first order, in: Selected Problems of Modern Continuum Theory, Bologne, June 3-6, 1987, W. Kosiński et al. (eds.), Pitagora, 1987, 195-205.
- [30] W. M. Zajączkowski, Global existence of solutions of noncharacteristic mixed problems to nonlinear symmetric dissipative systems of the first order, in: Proc. Second Conference on Hyperbolic Problems, March 1988; Notes Numer. Fluid Mech. 24 (1989), 688-697.
- [31] W. M. Zajączkowski, Existence and uniqueness of global in time solutions of mixed problems to nonlinear hyperbolic dissipative systems of the first order, Bull. Polish Acad. Sci. Math., to appear.
- [32] W. M. Zajączkowski, A priori estimates for solutions to noncharacteristic mixed problems to nonlinear symmetric hyperbolic systems of the first order with dissipation, ibid. 37 (1989), 184-197.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: Primary 35L45, 35L50
Identyfikator YADDA
bwmeta1.element.zamlynska-e1722792-fff3-4741-a5c7-3f3e5a16e3c0
Identyfikatory
ISBN
83-85116-17-6
ISSN
0012-3862
Kolekcja
DML-PL
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