November  2013, 33(11&12): 5293-5303. doi: 10.3934/dcds.2013.33.5293

How to distinguish a local semigroup from a global semigroup

1. 

Department of Mathematics, University of North Texas, Denton, TX 76205-5017, United States

Received  September 2011 Revised  March 2012 Published  May 2013

For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of `global' or `local' holds. Numerical examples are given. A linear transformation $A$ is defined so that one has `global' or `local' depending on whether $A$ does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems..
Citation: J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
References:
[1]

H. Brezis, "Operateurs Maximaux Monotones," North Holland, 1973. Google Scholar

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G. da Prato, "Applications Croissantes et Èquations d'évolutions dans les Espacè de Banach," Academic Press 1976.  Google Scholar

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J. R. Dorroh and J. W. Neuberger, A theory of strongly continuous semigroups in terms of lie generators, J. Functional Analysis, 136 (1996), 114-126. doi: 10.1006/jfan.1996.0023.  Google Scholar

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E. Hille and R. Phillips, "Functional Analysis and Semigroups," American Mathematical Society, 1957.  Google Scholar

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J. Goldstein, "Semigroups of Linear Operators and Applications," Oxford, 1985.  Google Scholar

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Sophus Lie, "Differential-Gleichungen," AMS Chelsea Publishing, 1967, (Originally published, Leipzig, in 1891). doi: 10.1007/BF01444840.  Google Scholar

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J. W. Neuberger, A generator for a set of functions, Ill. J. Math., 9 (1965), 31-39.  Google Scholar

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J. W. Neuberger, An exponential formula for one-parameter semigroups of nonlinear transformations, J. Math. Soc. Japan, 18 (1966), 154-157. doi: 10.2969/jmsj/01820154.  Google Scholar

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J. W. Neuberger, Lie generators for local semigroups, Contemporary Mathematics, 513 (2010), 233-238. doi: 10.1090/conm/513/10086.  Google Scholar

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J. W. Neuberger, "Sobolev Gradients and Differential Equations," Springer Lecture Notes in Mathematics 1670, 2010 (Second Edition). doi: 10.1007/978-3-642-04041-2.  Google Scholar

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J. W. Neuberger, "A Sequence of Problems on Semigroups," Springer Problem Books, 2011. doi: 10.1007/978-1-4614-0430-9.  Google Scholar

[12]

G. F. Webb, Representation of semigroups of nonlinear nonexpansive transformations in Banach spaces,, J. Math. Mech. 19 (1969/70), (): 159.   Google Scholar

show all references

References:
[1]

H. Brezis, "Operateurs Maximaux Monotones," North Holland, 1973. Google Scholar

[2]

G. da Prato, "Applications Croissantes et Èquations d'évolutions dans les Espacè de Banach," Academic Press 1976.  Google Scholar

[3]

J. R. Dorroh and J. W. Neuberger, A theory of strongly continuous semigroups in terms of lie generators, J. Functional Analysis, 136 (1996), 114-126. doi: 10.1006/jfan.1996.0023.  Google Scholar

[4]

E. Hille and R. Phillips, "Functional Analysis and Semigroups," American Mathematical Society, 1957.  Google Scholar

[5]

J. Goldstein, "Semigroups of Linear Operators and Applications," Oxford, 1985.  Google Scholar

[6]

Sophus Lie, "Differential-Gleichungen," AMS Chelsea Publishing, 1967, (Originally published, Leipzig, in 1891). doi: 10.1007/BF01444840.  Google Scholar

[7]

J. W. Neuberger, A generator for a set of functions, Ill. J. Math., 9 (1965), 31-39.  Google Scholar

[8]

J. W. Neuberger, An exponential formula for one-parameter semigroups of nonlinear transformations, J. Math. Soc. Japan, 18 (1966), 154-157. doi: 10.2969/jmsj/01820154.  Google Scholar

[9]

J. W. Neuberger, Lie generators for local semigroups, Contemporary Mathematics, 513 (2010), 233-238. doi: 10.1090/conm/513/10086.  Google Scholar

[10]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," Springer Lecture Notes in Mathematics 1670, 2010 (Second Edition). doi: 10.1007/978-3-642-04041-2.  Google Scholar

[11]

J. W. Neuberger, "A Sequence of Problems on Semigroups," Springer Problem Books, 2011. doi: 10.1007/978-1-4614-0430-9.  Google Scholar

[12]

G. F. Webb, Representation of semigroups of nonlinear nonexpansive transformations in Banach spaces,, J. Math. Mech. 19 (1969/70), (): 159.   Google Scholar

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