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Evolution equations with dynamic boundary conditions

Published online by Cambridge University Press:  14 November 2011

Thomas Hintermann
Thomas Hintermann
Affiliation:
Mathematisches Institut der Universität Zürich, Rämistrasse 74, CH-8001 Zürich, Switzerland
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Synopsis

In this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 43 - 60
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Agmon, S.. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure. Appl. Math. 15 (1962), 119–147.CrossRefGoogle Scholar
2Agmon, S., Douglis, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35–92.CrossRefGoogle Scholar
3Amann, H.. Gewöhnliche Differentialgleichungen (Berlin: Walter de Gruyter, 1983).Google Scholar
4Amann, H.. Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 593–676.Google Scholar
5Amann, H.. Semigroups and nonlinear evolution equations. Linear Algebra Appl. 84 (1986), 3–32.CrossRefGoogle Scholar
6Amann, H.. Dynamic theory of quasilinear parabolic equations, part I: abstract evolution equations. J. Nonlinear Anal.: T, M. and A. 12 (1988), 895–919.CrossRefGoogle Scholar
7Amann, H.. Dynamic theory of quasilinear parabolic equations, part II: reaction-diffusion-systems Diff. Int. Equations 3 (1990), 1–61.Google Scholar
8Bergh, J. and Löfström, J.. Interpolation Spaces. An Introduction (Berlin: Springer, 1976).CrossRefGoogle Scholar
9Courant, R. and Hilbert, D.. Methoden der Mathematischen Physik II, 2 (Berlin: Springer, 1968).CrossRefGoogle Scholar
10Diaz, J. I. and Jimenez, R. F.. Aplication de la teoria no lineal de semigrupos a un operador pseudodifferential. Actas V II C.E.D.Y.A. (1984), 137–142.Google Scholar
11Friedman, A. and Shinbrot, M.. The initial value problem for the linearized equations of water waves. J. Math. Mech. 17 (1968), 107–180.Google Scholar
12Garipov, R. M.. On the linear theory of gravity waves. Arch. Rational Mech. Anal. 24 (1967), 352–362.CrossRefGoogle Scholar
13Gröger, K.. Initial boundary value problems from semiconductor device theory. Z. Angew. Math. Mech. 67 (1987), 345–355.CrossRefGoogle Scholar
14Dalsen, M. Grobbelaar-Van. Semilinear evolution equations and fractional powers of a closed pair of operators. Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 101–115.CrossRefGoogle Scholar
15Hintermann, T.. Evolutionsgleichungen mit dynamischen Randbedingungen (Dissertation, Universität Zürich, 1988).Google Scholar
16Hörmander, L.. The analysis of linear partial differential operators I (Berlin: Springer, 1985).Google Scholar
17Lamb, H.. Hydrodynamics, 4th edn (Edinburgh: Cambridge University Press, 1916).Google Scholar
18Langer, R. E.. A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tôhoku Math. J. (2) 35 (1932), 260–275.Google Scholar
19Lions, J. L.. Quelques méthodes de résolutions des problèmes aux limites non linéaires (Paris: Dunod, 1969).Google Scholar
10Pazy, A.. Semigroups of linear operators and applications to partial differential equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
21Rossouw, W. J.. Behoudwetformulerings vir randvoorwaardes (Thesis, University of Pretoria, 1983).Google Scholar
22Sauer, N.. Linear evolution equations in two Banach spaces. Proc. Roy. Soc. Edinburgh Sect. A 91 (1982), 287–303.CrossRefGoogle Scholar
23Seeley, R.. Interpolation in L p with boundary conditions. Studia. Math. 44 (1972), 47–60.CrossRefGoogle Scholar
24Stahel, A.. Nichtlineare Randbedingungen bei Problemen vom Typ der Wellengleichung (Dissertation, Zürich, 1987).Google Scholar
25Triebel, H.. Interpolation theory, function spaces, differential operators (Amsterdam: North-Holland, 1978).Google Scholar
26Rensburg, N. F. J. Van. Dinamiese randvonvaardes vir parsiele differensiaalvergelykings (Thesis, University of Pretoria, 1982).Google Scholar
27Lewis, D., Marsden, J. and Ratiu, T.. Formal stability of liquid drops. In Perspectives in Nonlinear Dynamics, eds Schlesinger, L., Cawley, J. P., Saenz, R. and Zachary, W. W. pp. 71–83 (World Scientific, 1986).Google Scholar
28Okamoto, H.. Nonstationary or stationary free boundary problems for perfect fluid with surface tension. In Recent Topics in Nonlinear PDE, Hiroshima 1983, pp. 143–154 (Amsterdam: North-Holland, 1983).Google Scholar
29Benjamin, T. and Olver, P.. Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125 (1982), 137–185.CrossRefGoogle Scholar