# American Institute of Mathematical Sciences

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 - A, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
##### References:
 [1] H. Brezis, "Operateurs Maximaux Monotones,", North Holland, (1973). [2] G. da Prato, "Applications Croissantes et Èquations d'évolutions dans les Espacè de Banach,", Academic Press 1976., (1976). [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. doi: 10.1006/jfan.1996.0023. [4] E. Hille and R. Phillips, "Functional Analysis and Semigroups,", American Mathematical Society, (1957). [5] J. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford, (1985). [6] Sophus Lie, "Differential-Gleichungen,", AMS Chelsea Publishing, (1967). doi: 10.1007/BF01444840. [7] J. W. Neuberger, A generator for a set of functions,, Ill. J. Math., 9 (1965), 31. [8] J. W. Neuberger, An exponential formula for one-parameter semigroups of nonlinear transformations,, J. Math. Soc. Japan, 18 (1966), 154. doi: 10.2969/jmsj/01820154. [9] J. W. Neuberger, Lie generators for local semigroups,, Contemporary Mathematics, 513 (2010), 233. doi: 10.1090/conm/513/10086. [10] J. W. Neuberger, "Sobolev Gradients and Differential Equations,", Springer Lecture Notes in Mathematics 1670, (1670). doi: 10.1007/978-3-642-04041-2. [11] J. W. Neuberger, "A Sequence of Problems on Semigroups,", Springer Problem Books, (2011). doi: 10.1007/978-1-4614-0430-9. [12] G. F. Webb, Representation of semigroups of nonlinear nonexpansive transformations in Banach spaces,, J. Math. Mech. 19 (1969/70), (): 159.

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##### References:
 [1] H. Brezis, "Operateurs Maximaux Monotones,", North Holland, (1973). [2] G. da Prato, "Applications Croissantes et Èquations d'évolutions dans les Espacè de Banach,", Academic Press 1976., (1976). [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. doi: 10.1006/jfan.1996.0023. [4] E. Hille and R. Phillips, "Functional Analysis and Semigroups,", American Mathematical Society, (1957). [5] J. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford, (1985). [6] Sophus Lie, "Differential-Gleichungen,", AMS Chelsea Publishing, (1967). doi: 10.1007/BF01444840. [7] J. W. Neuberger, A generator for a set of functions,, Ill. J. Math., 9 (1965), 31. [8] J. W. Neuberger, An exponential formula for one-parameter semigroups of nonlinear transformations,, J. Math. Soc. Japan, 18 (1966), 154. doi: 10.2969/jmsj/01820154. [9] J. W. Neuberger, Lie generators for local semigroups,, Contemporary Mathematics, 513 (2010), 233. doi: 10.1090/conm/513/10086. [10] J. W. Neuberger, "Sobolev Gradients and Differential Equations,", Springer Lecture Notes in Mathematics 1670, (1670). doi: 10.1007/978-3-642-04041-2. [11] J. W. Neuberger, "A Sequence of Problems on Semigroups,", Springer Problem Books, (2011). doi: 10.1007/978-1-4614-0430-9. [12] G. F. Webb, Representation of semigroups of nonlinear nonexpansive transformations in Banach spaces,, J. Math. Mech. 19 (1969/70), (): 159.
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