Semigroup Generation

Transcription

1 Semigroup Generation Yudi Soeharyadi Analysis & Geometry Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung WIDE-Workshoop in Integral and Differensial Equations 2017 Institut Teknologi Bandung

2 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

3 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

4 From Previous Lectures The initial value problem or Abstract Cauchy Problem (ACP) d dt u = Au,u(0) = u 0 for some matrix (or operator in general) has solution u(t) = e t A u 0, t 0. Himpunan {e t A : t 0} disebut semigroup.

5 A Generalization Suppose X is a Hilbert (or Banach) space, a (linear) operator A : Dom(A) X X, and an initial value problem Some question: d dt u(t) = Au(t), u(0) = u 0 X. 1. When do we have solution, what are the conditions in term of A and X? 2. How do we represent a solution, if any? 3. In analogy to the case of finite dimensional (A is a square matrix), how do we make sense of e t A in a more general context?

6 Banach spaces A Banach space X is a linear space (vector space) over the complex (or real) field, equipped with norm, in which the norm gives "completeness" (i.e. every Cauchy sequence converges, in X ). With the norm, a metric can be defined, then the metric impose a topology (open-ness, closed-ness) to X. The imposed topology is complete. Examples of Banach spaces: Euclidean spaces R N,L p,l p Examples of non Banach space (not complete): C 0,C 1

7 Hilbert spaces A Hilbert space H is a Banach space, in which the norm is imposed by an inner product, that is a sesquilinear form, : H H F, such that x = x, x 1 2, for x H. Examples of Hilbert space: Euclidean space R N,l 2,L 2, H 1, H 2 Examples non Hilbert space: l p,l p for p 2, C 0,C 1

8 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

9 Semigroups Let X be a Banach (or a Hilbert) space. One parameter family of (bounded) operators {T (t) : X X : t 0} is said to be a strongly continuous (C 0 )- semigroup if 1. T (t) <, that is sup f X, f 1 T (t)f <, for each t T (0) = Id 3. T (t + s) = T (t)t (s) 4. The map t T (t)f is continuous for t 0 and f X. The semigroup is called C 0 -contraction if T (t) 1, for each t 0.

10 Generators Infinitesimal generator A of the semigroup is the operator T (h)f T (0)f A f := lim, h 0 + h and f is in the domain of A iff this limit exists.

11 An example Let T (t) bethe translation operator by t, that is Then T (h)f f lim h 0 + h T (t)f := f ( + t). = lim h 0 + f ( + h) f ( ), h that is the derivative f if the limit exists. Here A : f f. A = d dx. The domain should be a subset of the set of differentiable functions.

12 Remarks Formally, the definition of generator suggests that 1. T (t) = "e t A ", where A = d dt t=0t (t). 2. The solution to the ACP (1) is u(t) = T (t)u 0, where T (t) is the semigroup generated by A. So we say that the solution of the ACP (1) is obtained by evolving the initial data given by the semigroup generated by A.

13 Remarks Relation between infinitesimal generator and semigroup gives an intutive idea to solve a (time dependent) differential equation: 1. Rewrite the equation as an ACP, identify the operator A 2. Using A, generate the semigroup T (t) 3. The solution is u(t) = T (t)u 0. In fact this gives the following basic result.

14 Well-posedness Theorem. Given an Abstract Cauchy Problem (ACP) d dt u(t) = Au(t), u(0) = u 0, with a linear operator A. The ACP is well-posed iff A is the infinitesimal generator of C 0 -semigroup T (t). In this case, the unique solution of the ACP is given by u(t) = T (t)u 0, for u 0 in the domain of A.

15 Remarks For a nonhomogeneous ACP d dt u(t) = Au(t) + g (t), u(0) = u 0 X, Variation of Parameters formula (also called Duhamel Principle to get the solution u(t) = T (t)u 0 + t 0 T (t s)g (s)ds, t 0, here T (t) is the semigroup generated by A.

16 Question While it is quite clear how to get the infiinitesimal generator A from a given semigroup T (t) (using derivative at t = 0), it remains a mistery how the operator A generates the semigroup. More specifically, what kind of condition should be imposed to the operator A, so it becomes the infinitesimal generator of C 0 -semigroup T (t). This points to Semigroup Generation Problem

17 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

18 Resolvent Suppose T (t) is a C 0 -contraction semigroup, with generator A. Intuitively we consider T (t) = e t A. The formula 1 λ A = 0 e λt e t A dt, is true when A is a number, and λ > Re(A). This suggests the operator version (λi A) 1 f = 0 e λt T (t)f dt, is somehow true, although in some more restricted way. It turns out to be valid for all λ > 0, f X.

19 Resolvent The estimate 0 e λt T (t)f dt e λt T (t) dt 0 0 = f /λ. e λt f dt suggests that for each λ > 0 Cond1 λi A maps the domain of A onto X (surjection) Cond2 (λi A) 1 f f /λ, for all f X.

20 Hille-Yosida Theorem Theorem (Hille-Yosida generation theorem). A linear operator A generates a C 0 -contraction semigroup iff the domain of A is dense in X, and Cond1, Cond2 are satisfied. With this A, one can recover the semigroup using inverse Laplace transform, or using "exponential" formula ( T (t)f = lim I t ) n n n A f.

21 Remark Important implications of the previous theorems: A densely defined operator A satisfying Cond1 and Cond2, generates C 0 -semigroup. If A generates C 0 -semigroup T (t), then ACP is well posed and is governed by the semigroup. Cond1 is verified by solving equation of the form λh Ah = g and getting a solution h satsfying the estimate h g /λ, for any given g X and λ > 0.

22 Exercises 1. For the translation semigroup T (t)f = f ( + t), we see from previous example that the infinitesimal generator is A = d dx. Show that A really generates the translation semigroup, by showing Cod1 and Cond2. What is the appropriate X? 2. One (spatial) dimension first order transport equaton is a PDE of the form u t = cu x, u(x,0) = u 0 (x). It is known that the solution is u(x, t) = u 0 (x + ct). Reformulate the problem as an ACP, identify the space, and the operatora. Can you recognize the relation to Ex. 1?

23 An Extension Semigroup generation can be extended in the sense as in the following Theorem (Perturbation). If A generates a C 0 -semigroup and if B is a bounded linear operator on X, then A + B generates a C 0 -semigroups. Theorem (Approximation). Suppose A n generates a C 0 -semigroup contraction T n (t), for n 0. If Dom(A 0 ) Dom(A n ) and lim n A n f = A 0 f, for all f Dom(A 0 ), then lim (λi A n) 1 f = (λi A) 1 f, n for all λ > 0, f X and t > 0. lim T n (t)f = T 0 (t)f, n

24 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

25 Dissipative operators Let X be a Hilbert space with inner product,. and we le also that A is densely defined. The operator A is said to be dissipative if Re( A f, f ) 0, f Dom(A). It is said to be m-dissipative if A is dissipative and it satisfies range condition: cl(rang e(λi A)) = X, for λ > 0.

26 Lumer-Phillips Theorem Theorem (Hille-Yosida, Lumer-Phillips Form). The operator A generates a C 0 -contraction semigroup iff A is densely defined and m-dissipative.

27 Remarks According to Lumer-Phillips Theorem, to show semigroup generation we need: Choose a subset of X to be Dom(A) so that A is desnsely defined Show dissipativity: A f, f 0 Show range condition, that is solving λh Ah = g, for a given g X

28 Example: A f = f is m-dissipative. We choose X = L 2 (R), Dom(A) = {u X : u iscontnuous}, it is densely defined Au,u = u u dx = (u ) 2 dx 0, using integration by parts Range condition: Exercise

29 Exercises Formulate the following PDEs as abstract Cauchy problems. Are they well-posed? 1. Transport: u t = cu x 2. Heat: u t = u xx 3. Transport plus growth:u t = cu x + u 4. Heat plus quadratic growth: u t = u xx + u 2 5. Wave: u t t = u xx

30 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

31 Concluding remarks 1. We have shown the contexts where semigroup theory applies, although in somewhat restricted ways 2. The theory can be extended to nonlinear problems 3. Needs: linear algebra, ODE and PDE, lots of analysis: real, complex, measure and integration, functional analysis, Fourier analysis

32 Outline Introduction Strongly Continuous Semigroups Semigroup Generation Semigroup generation: Lumer-Phillips Form Conclusions References

33 References There is a comprehensive literature in the subject of semigroup of operators. On the subject we may suggest: 1. J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford A. Pazy, Semigroups of Operators and Its Applications, Springer-Verlag 3. K.J. Engel, R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS 1997