On the spectrum of the linear transport operator with generalized boundary conditions

Weak spectral mapping theorems forC0-groups associated to transport equations in slab geometry

“…The assumptions on ν(·,·) and i (·) are those of the previous section while κ(·, · ,·) is a measurable and non-negative 1]. Define the collision operator Recall that, usually in kinetic theory, the collision operator is an integral operator with respect to velocities and then induces some compactness with respect to velocities.…”

“…This is due mainly to the physical relevance of no-reentry boundary conditions in nuclear reactor theory [6]. We cite however some works dedicated to such models with general Maxwell-like boundary conditions, for example [1,3,25]. Recall that, in the kinetic theory of gases, Maxwell boundary conditions describe gas/surface interaction as a suitable balance of the incoming distribution of gas and the outgoing one.…”

Study of the Free Streaming Operator in Slab Geometry in Dependence of the Boundary Conditions

“…In [1,6] it is shown that the resolvent operator R(z, ¹ ) is defined for z'max(0, (v/2a) log ), "max( , ) and is a positive operator. Moreover, the operator ¹ is the generator of a strongly continuous semigroup of type )max(0, (v/2a) log ).…”

“…[1,4,8]) generates a strongly continuous contractive semigroup in an¸ space context, while conservative boundary conditions like specular reflection with restitution coefficient equal to one lead to a group of unitary norm. It is interesting to remark that the results known in the literature on the streaming operator with general boundary conditions were essentially based on their dissipativity up to a few years ago.…”

Semigroup generation properties of the streaming operator in dependence of the boundary conditions

“…IYlf(-a,Y;t) dyIff(')= A f'"', we have that 11 f(*);X(')Il 5 11 All 11 f(");X(")I( , hence + l + aI (1 yg ds)dy 2 11 f(");X(")II (1 -ll All , -1-awhere 11A11 is the norm of the bounded operator A, as defined in(10). Therefore, if llA\l 5 1 and z > 0, we obtain from (16):(1 g ( 1 2 ZII f II , )IR(z,TA)gll 5 &gll ; (19) ((exp(tT,)fII 5 llfll , vf E x, i.e.…”