**March 06, 2017 at 09:30 - March 10, 2017**
- BCAM

**Sébastien BRETEAUX, BCAM**

DATES: 6-10 March 2017 (7 sessions)

TIME: From Monday to Friday (10:00 - 11:30), on Monday & Wednesday (15:00 - 16:30). A total of 10.5 hours.

One of the fundamental problems in partial differential equations is the Cauchy problem u'(t)=Au(t), u(0)=x, with x in X, a Banach space of functions, and A a linear operator, e.g. the Laplace operator. In the simpler case when X is replaced by lR^d, the solution exp(tA)x of this problem is given using the exponential, and all the relevant information can be found either

– in the generator A,

– or in the exponential exp(tA), (semigroup viewpoint),

– or in the resolvent 1/(z-A) and the spectrum.

In a general Banach space, it is not clear whether exp(tA) makes sense. But the theory of “strongly continuous one-parameter semigroup” gives many results on such problems, and allow to relate the three viewpoints of generator, semigroup and the spectrum. The terminology of semigroup comes from an analogous of the “group” property exp((t+s)A) = exp(tA) exp(sA).

PROGRAMME

We will first present explicit examples of semigroups, and then we will turn to basic results of this theory, which proves the existence of the semigroup from hypothesis on the generator, and reciprocally how the generator can be extracted from the semigroup. We will also relate the semigroup and the generator to the resolvent.

Depending on time and the interest of the public we might also look at

– perturbations of semigroups,

– the Trotter-Kato theorem,

– dynamical systems: the more general case when A=A(t) depends on the parameter t.

– other subjects of interest to the public.

PREREQUISITES

– Basic knowledge in Functional Analysis. We will recall the standard functional analysis theorems we use, without proof.

***Registration is free, but inscription is required before 1st March: **So as to inscribe send an e-mail to

roldan@bcamath.org. Student grants are available. Please, let us know if you need support for travel and accommodation expenses.