Quasilinear parabolic problems with nonlinear boundary conditions [Elektronische Ressource] / von Rico Zacher

Sujets

• Mathematics

Informations

• Publié par: martin-luther-universitat_halle-wittenberg
• Publié le: 01 janvier 2003
• Nombre de lectures: 20
• Langue: English

Extrait

Quasilinear parabolic problems with
nonlinear boundary conditions
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
vorgelegt der
Mathematisch-Naturwissenschaftlich-Technischen Fakult¨at
(mathematisch-naturwissenschaftlicher Bereich)
der Martin-Luther-Universit¨at Halle-Wittenberg
von
Herrn Dipl.-Math. Rico Zacher
geb. am: 04. Dezember 1973 in: Weimar
Gutachter:
1. Prof. Dr. Jan Pruß,¨ Halle (Saale)
2. Prof. Dr. Philippe P. J. E. Cl´ement, Delft
3. Prof. Dr. Hana Petzeltova,` Prag
Halle (Saale), 14. Februar 2003 (Tag der Verteidigung)
urn:nbn:de:gbv:3-000004744
[ http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000004744 ]To My ParentsContents
1 Introduction 3
2 Preliminaries 11
2.1 Some notation, function spaces, Laplace transform . . . . . . . . . . . . . 11
2.2 Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Sums of closed linear operators . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Joint functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Operator-valued Fourier multipliers . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Evolutionary integral equations . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Volterra operators in L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27p
3 Maximal Regularity for Abstract Equations 33
3.1 Abstract parabolic Volterra equations . . . . . . . . . . . . . . . . . . . . 33
3.2 A general trace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 More time regularity for Volterra equations . . . . . . . . . . . . . . . . . 46
3.4 Abstract equations of first and second order on the halfline . . . . . . . . 47
3.5 Parabolic Volterra equations on an infinite strip . . . . . . . . . . . . . . . 48
4 Linear Problems of Second Order 57
4.1 Full space problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Half space problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Pointwise multiplication . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.3 Variable coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Problems in domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Linear Viscoelasticity 79
5.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Assumptions on the kernels and formulation of the goal . . . . . . . . . . 81
5.3 A homogeneous and isotropic material in a half space . . . . . . . . . . . 82
5.3.1 The case δ ≤δ : necessary conditions . . . . . . . . . . . . . . . . 83a b
5.3.2 The case δ ≤δ : sufficiency of (N1) . . . . . . . . . . . . . . . . . 84a b
5.3.3 The case 0<δ −δ <1/p . . . . . . . . . . . . . . . . . . . . . . 91a b
5.3.4 The case δ −δ >1/p . . . . . . . . . . . . . . . . . . . . . . . . . 92a b
16 Nonlinear Problems 95
6.1 Quasilinear problems of second order with nonlinear boundary conditions 95
6.2 Nemytskij operators for various function spaces . . . . . . . . . . . . . . . 102
Bibliography 111
2Chapter 1
Introduction
The present thesis is devoted to the study of the L -theory of a class of quasilinearp
parabolic problems with nonlinear boundary conditions. The main objective here is to
prove existence and uniqueness of local (in time) strong solutions of these problems. To
achieve this we establish optimal regularity estimates of typeL for an associated linearp
problem which allow us to reformulate the original problem as a fixed point equation
in the desired regularity class, and we show that under appropriate assumptions the
contraction mapping principle is applicable, provided the time-interval is sufficiently
small.
We describe now the class of equations to be studied. Let Ω be a bounded domain
n 2inR with C -smooth boundary Γ which decomposes according to Γ = Γ ∪Γ withD N
dist(Γ ,Γ ) > 0. For the unknown scalar function u :R ×Ω →R, we consider the+D N
subsequent problem:

2
 ∂ u+dk∗(A(u):∇ u)=F(u)+dk∗G(u), t≥0, x∈Ωt
B (u)=0, t≥0, x∈ΓD D (1.1)
 B (u)=0, t≥0, x∈ΓN N
u| =u , x∈Ω.t=0 0
Rt
Here,(dk∗w)(t,x)= dk(τ)w(t−τ,x), t≥0, x∈Ω,∂ umeansthepartialderivativeoft0
2uw.r.t. t,∇u=∇ uisthegradientofuw.r.t. thespatialvariables,∇ udenotesitsHes-x Pn2sian matrix, that is (∇ u) = ∂ ∂ u, i,j ∈{1,...,n}, and B : C = B Cij x x ij iji j i=1,j=1
n×nstands for the double scalar product of two matrices B, C ∈R . Furthermore, we
have the substitution operators
A(u)(t,x)=−a(t,x,u(t,x),∇u(t,x)), t≥0, x∈Ω,
F(u)(t,x)=f(t,x,u(t,x),∇u(t,x)), t≥0, x∈Ω,
G(u)(t,x)=g(t,x,u(t,x),∇u(t,x)), t≥0, x∈Ω,
DB (u)(t,x)=b (t,x,u(t,x)), t≥0, x∈Γ ,D D
NB (u)(t,x)=b (t,x,u(t,x),∇u(t,x)), t≥0, x∈Γ ,N N
n×n D Nwhere a isR -valued, and f, g, b , b are all scalar functions. The scalar-valued
kernel k is of bounded variation on each compact interval [0,T] with k(0) = 0, and
belongs to a certain kernel class with parameter α ∈ [0,1) which contains, roughly
αspeaking, all ’regular’ kernels that behave like t for t(> 0) near zero. Note that this
3formulation includes the special case k(t) = 1, t > 0, in which (1.1) amounts to the
quasilinear initial-boundary value problem

2∂ u+A(u):∇ u=H(u), t≥0, x∈Ω t
B (u)=0, t≥0, x∈ΓD D
(1.2)
B (u)=0, t≥0, x∈Γ N N

u| =u , x∈Ω,t=0 0
where H(u) = F(u) +G(u). Observe further that the case k(t) = t, t ≥ 0, is not
admissible; in our setting, this kernel would lead to a hyperbolic problem.
Although there is a wide literature on problems of the form (1.1), not much seems to
be known towards an L -theory in the integrodifferential case with nonlinear boundaryp
conditions, even in the linear situation with inhomogeneous Dirichlet and/or Neumann
boundary conditions. Before presenting the main result concerning (1.1) and comment-
ing on available results in the literature we give some motivation for the study of these
problems.
Equations of the form (1.1) appear in a variety of applied problems. They typically
arise in mathematical physics by some constitutive laws pertaining to materials with
memory when combined with the usual conservation laws such as balance of energy or
balance of momentum. To illustrate this point, we give an example from the theory of
heatconductionwithmemory. Fordetailsconcerningtheunderlyingphysicalprinciples,
werefertoNohel[59]. SeealsoCl´ementandNohel[23],Cl´ementandPru¨ss[25],Lunardi
[54], Nunziato [60], and Pru¨ss [63] for work on this subject.
Example: (Nonlinear heat flow in a material with memory)
Consider the heat conduction in a 3-dimensional rigid body which is represented by a
3 1boundeddomainΩ⊂R withboundary∂ΩofclassC . Letε(t,x)denotethedensityof
internalenergyattimet∈Randpositionx∈Ω,q(t,x)theheatfluxvectorfield,u(t,x)
the temperature, andh(t,x)theexternalheatsupply. Thelawofbalanceofenergythen
reads as
∂ ε(t,x)+divq(t,x)=h(t,x), t∈R, x∈Ω. (1.3)t
Equation(1.3)hastobesupplementedbyboundaryconditions;thesearebasicallyeither
prescribed temperature or prescribed heat flux through the boundary, that is to say
u(t,x) = u (t,x), t∈R, x∈Γ , (1.4)b b
−q(t,x)·n(x) = q (t,x), t∈R, x∈Γ , (1.5)f f
where Γ and Γ are assumed to be disjoint closed subsets of ∂Ω with Γ ∪Γ = ∂Ω,b f b f
and n(x) denotes the outer normal of Ω at x∈∂Ω. In order to complete the system we
havetoaddconstitutiveequationsfortheinternalenergyandtheheatfluxreflectingthe
properties of the material the body is made of. In what is to follow we shall consider an
isotropic and homogeneous material with memory. Following [23], [39], [60] and many
other authors, we will use the laws
Z ∞
ε(t,x) = dm(τ)u(t−τ,x), t∈R, x∈Ω, (1.6)
0
Z ∞
q(t,x) = − dc(τ)σ(∇u(t−τ,x)), t∈R, x∈Ω, (1.7)
0
1 3 3wherem, c∈BV (R ),andσ∈C (R ,R )aregivenfunctions. Notethattheheatfluxloc +
here depends nonlinearly on the history of the gradient of u. It is physically reasonable
4to assume that m, c are bounded functions of the form m(t) =m +(1∗m )(t), t> 0,0 1
m(0) = 0, and c(t) =c +(1∗c )(t), t> 0, c(0) = 0, respectively, with m > 0, c ≥ 0,0 1 0 0
and m , c ∈ L (R ). Here and in the sequel, f ∗f denotes the convolution of two1 1 1 + 1 2
Rt
functions defined by (f ∗f )(t)= f (t−τ)f (τ)dτ, t≥0.1 2 1 20
Without loss of generality we may assume that the material is at zero temperature
uptotimet=0,andisthenexposedtoasuddenchangeoftemperatureu(0,x)=u (x),0
x ∈ Ω; otherwise one has to add a known forcing term in both (1.8) and (1.10) below
that incorporates the historyof the temperature up to timet=0. Then(1.3)-(1.7) yield
∂ (dm∗u)−dc∗(divσ(∇u)) = h, t>0, x∈Ω, (1.8)t
u = u , t>0, x∈Γ , (1.9)b b
dc∗σ(∇u) = q , t>0, x∈Γ , (1.10)f f
u| = u , x∈Ω. (1.11)t=0 0
We show now that (1.8)-(1.11) can be transformed to a problem of the form (1.1),
see also [23]. Note first that without restriction of generality we may assume m = 1.0
By integrating (1.8) with respect to time we obtain
u+m ∗u−c∗(divσ(∇u))=1∗h+u , t≥0, x∈Ω. (1.12)1 0
Define the resolvent kernel r∈L (R ) associated with m as the unique solution of1,loc + 1
the