a necessary and sufficient condition for global existence ......a necessary and sufficient...
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A Necessary and Sufficient Condition forGlobal Existence for Nonlinear Semigroups
J. W. Neuberger
University of North Texas
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Global Semigroups:
X : Polish Space.T (t) : X → X , t ∈ [0,∞).T (0)x = x , x ∈ X .T (t)T (s) = T (t + s), t , s ≥ 0.T jointly continuous (ifg : [0,∞)× X → X so thatg(t , x) = T (t)x , t ≥ 0, x ∈ X then gis continuous.)
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Conventional Generator for T :
X : subset of a Banach space YT : a semigroup on XB = {(x , y) ∈ X × Y :y = limt→0+ 1t (T (t)x − x) }B called conventional generator of T .
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Lie Generator for T on X :
CB(X ) : B-space of all boundedcontinuous functions X → R.A = {(f ,g) ∈ CB(X )2 :
g(x) = limt→0+
1t(f (T (t)x)−f (x)), x ∈ X}.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Local Semigroup T :
m, continuous X → (0,∞],x ∈ D(T (t)) ⇐⇒ t ∈ [0,m(x)).If t , s ≥ 0, x ∈ X , then T (t)T (s)x =T (t + s)x ⇐⇒ t + s < m(x).T jointly continuous, maximal(limt→s−T (t)x exists =⇒ s < m(x))
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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A Word from Sophus Lie:‘....all problems related tothe one-parameter groupmay be solved by means ofthe infinitesimaltransformation of the group.’
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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‘.....alle auf die eingliedrigeGruppe bezüglichen Problemedurch Benutzung derinfinitesimalen Transformationderselben allein gelöst werdenkönnen.’
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Generators and Semigroups:
Hille-Yosida Theorem characterizesstrongly continuous linearsemigroups in terms of theirconventional generators.Dorroh-N Theorem characterizesnonlinear jointly continuoussemigroups on X in terms of their Liegenerators.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Definitions
CB(X ): Bounded, continuous realfunctions on X .β convergence on X : Uniformconvergence on compacta.SG(X ): Global semigroups on X .LG(X ): Linear derivations, β densedomain, nonexpansive resolventswith equicontinuity property.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Theorem (Dorroh-N):If A ∈ LG(X ), ) there is a uniqueT ∈ SG(X ) with Lie generator A.Moreover
f (T (t)x) = limn→∞
((I − tn
A)−nf )(x),
x ∈ X , t ≥ 0, f ∈ CB(X ).
Conversely, if T ∈ SG(X ) and A its Liegenerator, then A ∈ LG(X ).
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Example of a Local Semigroup:
X = [0,∞).T (t)x = x1−tx , t large as possible,x ≥ 0.Generated by solutions z toz(0) = x , z ′(t) = z(t)2, t large aspossible.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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When is a Semigroup Local, Global?
Theorem: SupposeT is either a local or global jointlycontinuous semigroup andA is the Lie generator of T .
Then, A has a positive eigenvalue, witheigenfunction in CB(X ), if and only if Tis local.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Moreover . . . . . .If f (x) = exp(−m(x)), x ∈ X ,
then f ∈ CB(X ).f is an eigenvector of Aif and only if T is local(m(x)
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Example for Theorem:
For X = [0,∞),B(x) = x2, x ∈ X ,corresponding semigroup T is givenby T (t)x = x1−tx , x ≥ 0, t ∈ [0,
1x ).
Eigenvector of Lie generator A is f :
f (x) = exp(−1x), x > 0, f (0) = 0.
1 is an eigenvalue of A.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Characterization of Eigenfunctions of A:
Theorem: Suppose that T is a localsemigroup on X , A is the Lie generatorof T and
f (x) = exp(−m(x)), x ∈ X . (1)
If g ∈ CB(X ), Ag = g and x ∈ X , thenthere is c ∈ R so that
g(T (t)x) = cf (T (t)x), t ∈ [0,m(x)).
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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A Numerical Attack:
Compute spectra of discrete versionsof Lie generators.Use discrete versions of Sobolevspaces X and of CB(X ).Single ODE examples.Pairs of ODEs
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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Some Test Cases in One Dimension:
u′ = u2
u′ = u(u − 1)(u − 2)u′ = uu′ = −u2
u′ = exp(u/10)u′ = u3 − u
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 2 4 6 8 10 12 14 16 18 20−1
0
1
2
3
4
5
6
7
8n=1000, b(u) = u2
Figure: u′ = u2
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 2 4 6 8 10 12 14 16 18 20−20
0
20
40
60
80
100n=2000, b(u) = u*(u−1)*(u−2)
Figure: u′ = u(u − 1)(u − 2)
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 2 4 6 8 10 12 14 16 18 20−0.2
0
0.2
0.4
0.6
0.8
1
1.2n=2000, b(u) = u
Figure: u′ = u
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 2 4 6 8 10 12 14 16 18 20−6
−5
−4
−3
−2
−1
0
1x 10−8 n=10000, b(u) = −u
2
Figure: u′ = −u2
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 20 40 60 80 1000
5
10
15
20
25
30
35
40
45n=20000, b(u) = exp(u/10)
Figure: u′ = exp(u/10)
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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0 5 10 15 20−20
0
20
40
60
80
100
120n=2000, b(u) = u3 − u
Figure: u′ = u3 − u
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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A Decoupled System of Two ODEs
u(0) = x ≥ 0, u′ = u2,v(0) = y ≥ 0, v ′ = v2.
Observe
m(x , y) = min(1x,1y) x , y ≥ 0.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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02
46
810
0
2
4
6
8
10−0.01
0
0.01
0.02
0.03
0.04
Figure: u′ = u2, v ′ = v2
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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A Second Decoupled Pair
u(0) = x ≥ 0, u′ = u2,v(0) = y ≥ 0, v ′ = −v2.
Observe
m(x , y) =1x, x , y ≥ 0.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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02
46
810
0
2
4
6
8
10−0.01
0
0.01
0.02
0.03
0.04
0.05
Figure: u′ = u2, v ′ = −v2
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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A Coupled System
u(0) = x ≥ 0, u′ = v2,v(0) = y ≥ 0, v ′ = u2.
An expression for the correspondingstoping time function is not available,but it can be shown that the underlyingsemigroup is local.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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02
46
810
0
2
4
6
8
10−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Figure: u′ = v2, v ′ = u2
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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John M. Neuberger, Jim Swiftand Nandor Sieben are collaborating onnumerics in more than one dimension.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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References:
J. R. Dorroh, J. W. Neuberger, A Theoryof Strongly Continuous Semigroups inTerms of Lie Generators, J. FunctionalAnalysis, 136 (1996), 114-126.J. W. Neuberger, Lie Generators forLocal Semigroups, ContemporaryMathematics, 513 (2010), 233-238.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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J. W. Neuberger, How to distinguishlocal semigroups from globalsemigroups, Discrete amd ContinuousDynamical Systems-A, 33, (2012),5293-5303.Preprint at arXiv.org/abs/1109.2184.J. W. Neuberger, A Sequence ofProblems on Semigroups, SpringerProblem Book Series, (2012).
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups
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John M. Neuberger, J. W. Neuberger,James W. Swift, A Linear ConditionDetermining Local or Global Existencefor Nonlinear Problems, CentralEuropean J. Mathematics, 11 (2013),1361-1374.
J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups