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Partial integration in Rn
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Title page
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Partial integration on surfaces
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
Divergence type equations
[chap:DifferentialEquations]
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This is an english version of the script
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You may switch to the original german version:
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Many important differential operators have a divergence structure, e. g.
Here div is the "divergence operator".
In this chapter we shall study two special classes
of differential equations with divergence structure:
- EULER-LAGRANGE equations.
They arise as first variation of functionals
(see satz:7-3).
- Conservation laws.
They are the basic equations in continuum physics,
therefore present in most areas of natural science
(see satz:7-8 and satz:7-11).
The reason for the divergence structure, as we shall see,
in both cases is the divergence theorem
(see satz:6-7).
First let us consider functionals,
that is mappings
where we want X to be a function space.
The classical example is the space
X= C1(clos(Ω);RN) ,
and the case that E(u) for u∈M:=X
is an integral over Ω , in which values
and first derivatives of the function u enter:
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| Zum Minimumproblem [fig:min-1] |
In order to characterize minima of functionals
E:M⊂X → R ,
we need a notion of a "derivative" for E .
Such a notion goes beyond the well known notion in EUKLIDean
spaces (introduced in Analysis II).
The reason is, that here in general
- M is infinite dimensional,
- M is not a subspace.
In our general setting the "derivative"
of a mapping E:M⊂X → R at a point
u∈M will be obtained
by computing all "directional derivatives" ∂v E(u)∈R ,
provided they exist.
Here v runs over the "tangent cone" Tu(M) .
(Both notions will be introduced in definition:7-2.)
If u is an absolute (or local) minimum von E on M ,
one obtaines properties for these directional derivatives,
which depend on the position of u within M :
Now comes the general definition of
tangent cones (resp. tangent spaces) and directional derivatives.
An application of this definition to functionals
E:C1(clos(Ω);RN) → R as in definition:7-1
leads to the "first variation" of E .
Applying the divergence theorem for the integral in the
first variation one obtains the
Now we present some important special cases for the functional
in definition:7-1.
Beispiele [sect:7-6]
- [sect:7-6-(1)]
DIRICHLET integral type functionals.
Consider the scalar case ( N=1 ) and let
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E(u) :=
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- [sect:7-6-(2)]
Arc length functional.
Let γ be a parametrization of the curve Γ⊂Rn
and assume that γ∈M , where M is defined by
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u′(t)≠0
for t∈[t0,t1] and u injective}.
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As functional on M consider the arc length
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E(u) := H1(u([t0,t1]))
=
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|u′(t)| dt .
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- [sect:7-6-(3)]
Surface area functional.
Let Ω⊂Rn-1 be open and bounded,
u0∈C1(clos(Ω);R) and
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M:={u ∈C1(clos(Ω);R); u=u0 on ∂Ω}.
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As functional on M consider the surface area
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E(u) := Hn-1( graph (u))
=
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sqrt(1+|∇u(y)|2) dy .
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In the special case of one dimensional problems the following holds:
Theorem [satz:7-7]
Let n=1 and I⊂R an open interval.
Moreover, let N∈N , and
f:RN×RN → R twice continuously differentiable and
(that is, n=1 and f independent of the variable x (!)).
The the following holds:
If u∈C2(clos(I);RN) satisfies the
EULER-LAGRANGE equation
then
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u′•f′p(u,u′) - f(u,u′) = const.
on I .
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In the second part of this chapter we treat conservation laws.
Physically, such laws are described as follows:
Let q:R3 → R3 be a flux density of a scalar quantity
(e. g. density, concentration, temperature, energie etc.).
The main assumption of Continuum Physics states
(here we consider the stationary, that is, the time-independent case):
If V⊂R3 is an open bounded set
( V is a test volume),
then the gain of the physical quantity in V
corresponds to the loss of this quantity on ∂V :
Here f denotes the source density,
and νV denotes (as usual) the outer normal of V .
The above itendity also is called conservation law
with respect to the flux q .
We now present equivalent formulations of this property.
Satz (Divergence equation) [satz:7-8]
Let D⊂Rn be an open set,
q:D → Rn a vector field, and f:D → R a function.
Consider the following three properties:
- [satz:7-8-(1)]
Formulation as differential equation.
q∈C1(D;Rn) , f∈C0(D;R) , and
- [satz:7-8-(2)]
Formulation with test volumina.
q∈C0(D;Rn) , f∈L1(D;R) , and for all
bounded GAUSS domains V
(as defines in definition:6-7)
with clos(V)⊂D
(Alternatively, it suffices to consider this statement
for all balls V with clos(V)⊂D .)
- [satz:7-8-(3)]
Formulation with test functions.
q∈L1(D;Rn) , f∈L1(D;R) , and for all
ζ∈C0∞(D;R)
Then the following holds:
Each two of these statements are equivalent
under the corresponding stronger assumption
for q and f .
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It is common in many physical situations,
that there are solutions of the
divergence equation div (q)=f ,
which are not smooth at the interface between two media.
These solutions belong to a spezial class of weak solutions:
The corresponding flux satisfies a certain jump condition
at the interface.
Theorem (Jump condition) [satz:7-10]
Let Ω=Ω(1)∪Γ∪Ω(2)
be an open connected set, where
Ω(l) are open disjoint sets for l=1,2 , and
Γ= Ω∩∂Ω(1)=Ω∩∂Ω(2)
a C1 -surface.
(Thus Ω is decomposed in two open sets
Ω(l) for l=1,2 and the common boundary Γ .)
Further, let
q(l)∈C1(clos(Ω(l));Rn) and
f(l)∈C0(clos(Ω(l));R) for l=1,2 ,
and set
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q(x) := q(l)(x) for x∈Ω(l)
and
f(x):=f(l)(x) for x∈Ω(l) .
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Then q and f are integrable,
that is q∈L1(Ω;Rn) and f∈L1(Ω;R) .
Moreover, the following holds:
(q,f) is a wweak solution
of the differential equation div (q)=f in Ω ,
if and only if:
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[eq:7-differential-equation] |
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div (q(l))=f(l)
in Ω(l) for l=1,2,
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and
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q(l)•νΩ(l) = 0
on Γ (Jump condition).
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Notice:
It is νΩ(1)= - νΩ(2) on Γ .
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Remark:
The jump condition implies (q(1)-q(2))•ν=0 on Γ
for every normal field ν
(that is, ν(x)∈Tx(Γ)⊥ for x∈Γ ,
or equivalently,
q(1)(x)-q(2)(x)∈Tx(Γ) for x∈Γ ).
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Now let us consider the time dependent case:
The coordinates are (t,x)∈R×Rn ,
where the physical case for the space dimension is n=3 .
Let
- (t,x) |→ e(t,x) a scalar physical quantity,
- (t,x) |→ q(t,x) the corresponding flux density,
- (t,x) |→ f(t,x) the corresponding source density.
We consider the conservation law
In the following let us denote
by div the spatial divergence operator in Rn
and by div' the time-space divergence operator
in R1+n=R×Rn .
With this convention the conservation law becomes
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∂te + div (q) = div '(e,q) = f
.
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We call this the conservation law
for the physical quantity e .
This is the fundamental differential equation in Continuum Physics.
In physics literature it is usually stated
as property for test volumina
V⊂Rn :
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e(t,x) dx
=
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f(t,x) dx
-
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q(t,x)•νV(x) dHn-1
.
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The purpose of the following theorem is to prove,
that these two formulations are equivalent.
Theorem (Conservation law) [satz:7-11]
Let D = I ×Ω⊂R×Rn be an open set,
with coordinates t ∈I (e. g. the time)
and x ∈Ω (e. g. the position).
Moreover, let e,qk,f : D → R for k=1,...,n .
Consider the following properties:
- [satz:7-11-(1)]
Formulation as differential equation.
Let e,qk ∈C1(D) , f ∈C0(D) ,
and assume, that in I ×Ω
- [satz:7-11-(2a)]
Formulation with test volumina in Ω .
Let e∈C1(D) , qk,f ∈C0 (D) ,
and assume, that for t∈I
and all open sets V ⊂Ω as in satz:7-8
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e(t,x) dx
+
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q(t,x) •νV (x) dHn-1 (x)
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- [satz:7-11-(2b)]
Formulation with test volumina in I ×Ω .
Let e,qk ∈C0(D) , f ∈L1 (D) ,
and assume, that for all [t1 ,t2] ×V ⊂D
with V as in satz:7-8
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e(t,x) dx
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+
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q(t,x)
•νV (x) dHn-1 (x) dt
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- [satz:7-11-(3a)]
Formulation with test functions in Ω .
Let e ∈C1(D) , qk,f ∈L1(D) ,
and assume, that for all η∈C0∞(Ω)
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η(x) e(t,x) dx
=
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(∇η(x) •q(t,x) + η(x) f(t,x)) dx .
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- [satz:7-11-(3b)]
Formulation with test functions in I ×Ω .
Let e,qk,f ∈L1(D) ,
and assume, that for all ζ∈C0∞( I ×Ω)
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( ∂tζ·e
+ ∇ζ•q + ζ·f ) dLn+1 = 0 .
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Then the following holds:
Each two of these statements are equivalent
under the corresponding stronger assumption
for e , q , and f .
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Rest of this part has not yet been translated
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The following is the original german version
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Das wichtigste Beispiel eines Systems von Erhaltungsgleichungen
in der Hydrodynamik ist das
Version 1.7
H.W. Alt - 02.01.2007
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Partial integration in Rn
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Title page
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Partial integration on surfaces
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany