We consider the initial boundary value problem for nonlinear degenerate parabolic equations of the form

Theory of BV functions and its application to nonlinear degenerate parabolic equations

Department of Mathematics Graduate School of Science and Engineering Chuo University

Hiroshi Watanabe

We consider the initial boundary value problem for nonlinear degenerate parabolic equations of the form

(CP) 8 >

> >

> <

> >

> >

:

u

+ r Å A(x; t; u) + B(x; t; u) = Åå (u); (x; t) 2 ä Ç (0; T );

@u

@ n (x; t) = 0; (x; t) 2 @ä Ç (0; T );

u(x; 0) = v (x); v 2 L

(ä); given:

Here r = (@=@x

; : : : ; @=@x

) and Å = P

i=1

@

=@x

are the spatial nabla and the Laplacian in R

, respectively, and [0,T] is a åxed time interval. ä is a rectangle with sides are parallel to the coordinate axes in R

. A(x; t; ò ) = (A

; :::; A

)(x; t; ò ) is an R

-valued diãerentiable function on ä Ç [0; T ] Ç R and B(x; t; ò ) is an R -valued diãerentiable function on ä Ç [0; T ] Ç R . The function å on the right-hand side is supposed to be strictly increasing and locally Lipschitz continuous on R . n represents the unit normal to @ä. In this paper we årst outline the main point of the theory of functions of bounded variation, and then awake an attempt to construct weak solutions to (CP) along Okamoto and Oharu [3] which deals with the initial value problem to (CP) over the whole space R

. They consider a Lax- Friedrichs type diãerence approximation to (CP) in the Frechet space L

( R

) and apply an approximation theory for nonlinear evolution operator to construct solution operator fU (t; s) ; 0 î s î t î T g on L

( R

), viewed as a subspace of L

( R

), which provide weak solutions to (CP).

We want to rewrite this result over the bounded domain under the Neumann boundary condition. To this end, we must overcome the crucial problem. First, we discuss certain technical diéculties which arise when we employ diãerence operators. In this paper we are particularly interested in the homogeneous Neumann boundary condition on the boundary

@ä. The reason why we treat this situation is due to discrete methods in numerical analysis.

In numerical analysis, Neumann boundary condition is treated in such a way that numerical solutions are supposed to take the same values on the discretized points near the boundary.

This method is used extensively in standard computing. Here we aim to make a mathematical description of this situation.

Function spaces.

We wish to treat the initial-boundary value problem in L

(ä) and the solutions lying in BV (ä). We devide the class L

(ä) of initial data in (CP) into a family of absolutely convex subsets deåned by

(1) X

= f v 2 L

(ä) ; jj v jj

î r g for r > 0:

Abstract Cauchy Problems.

Deånition 1

Let t 2 [0; T ]. We say that v 2 ( A (t)) and w 2 A (t)v, if v 2 L

(ä) \ BV (ä); w 2 L

(ä) and

h å (v); Å' i + h A( Å ; t; v); r ' i Ä h B ( Å ; t; v); ' i = h w; ' i for all ' 2 C

(ä)

Now the problem (CP) is converted to a time-dependent abstract Cauchy problem in L

(ä) of the form

(ACP)

( (d=dt)u(t) = A (t)u(t) for t 2 (0; T );

u(0) = v 2 L

(ä);

where (d=dt)u(t) is understood to be the derivative of u( Å ) with respect to t (perhaps in a generalized sense).

Diãerence approximation.

We employ a diãerence approximation to the problem (CP). Let ` > 0 be a mesh size of spatial diãerence. Then

(DS) 8 >

> >

> >

<

> >

> >

> :

u

= v 2 L

(ä);

u

Ä u

h +

X

D

(`)A

( Å ; jh; u

) + D

(`)B( Å ; jh; u

) = X

i=1

D

(`)D

(`)å (u

);

h > 0; 0 î j î [T =h]:

Here D

(`)v = `

[v( Å + `e

) Ä v]; D

(`)v = `

[v Ä v( ÅÄ `e

)];

D

(`)v = (2`)

[v ( Å + `e

) Ä v ( ÅÄ `e

)]; D

(`)v = (2N )

P

i=1

[v( Å + `e

) + v( ÅÄ `e

)]:

At this point, we impose the CFL conditon. If v 2 X

in (DS), then h 2 (0; h

) ; é

= h=`

; "

(h) = M

(h=é

)

= M

`:

(2)

Here, r 2 (0; 1 ) 7! é

2 (0; 1) is a monotone nonincreasing function such that (3) 0 < é

< min

ö 1

2N K

; 1 N M

õ

; and h

= min ( !

3 ; é

í K

M

í 1

2N K

Ä é

ì ì

) : In view of (2), we deåne the diãerence operators C

(t) : X

! L

(ä); r > 0; t 2 [0; T ];

h 2 (0; h

); by

(4)

C

(t)v = v + h

" X

i=1

D

(`)D

(`)(å (v) + "

(h)v ) Ä

X

D

(`)A

( Å ; t; v) Ä D

(`)B( Å ; t; v)

#

for v 2 X

.

Boundary condition.

We now put (v ( ÅÜ í`e

)) = ä

; and ä

= S

i

(ä Ä ä

); ä

= S

i

(ä

Ä ä);

ä

= S

i

(ä

Ä ä) for 0 î í î 1: We consider a Neumann boundary condition for the solutions. At this point we assume that the domain so that solutions can be also extended, in such a way that the extended solutions satisfy the homogeneous Neumann condition hold.

We employ a class of initial data which consists of functions v such that (5)

Z

äíÜ Ü

vdx Ä Z

äíÜ

vdx = 0:

In later arguments, we may employ the following condition : (6)

Z

äíÜ Ü

vdx Ä Z

äíá

vdx î `c Z

ä

vdx:

Product formulae.

We deåne (

( A

(t)) = X

A

(t) = h

[ C

(t) Ä I] for r > 0; t 2 [0; T ]; and h 2 (0; h

):

The diãerential operator A

(t) approximates the diãerential operator A (t) formulated in Deånition 1.1 as h # 0 in the following sense:

(I Ä ï A

(t))

v ! (I Ä ï A (t))

v in L

(ä) as h # 0

for ï> 0 with 0 < ï< min f ã

; ã

(h

)

g . Here, ã

(h) = (e

Ä 1)=h. We denote fU

(t; s) ; h 2 (0; h

) g by the family of solution operators generated by the family of diãerence operators fA

(t) ; h 2 (0; h

); t 2 [0; T ] g .

Also, the symbols X

(s); R > 0 s 2 [0; T ]; denote absolutely convex subsets of L

(ä) deåned by

X

(s) = f v 2 L

(ä) j G

( jj v jj

) î R g :

Here the family of positive functionals f G

( Å ) ; s 2 [0; T ] g is chosen so that the class of initial data may be classiåed in terms of the values of G

( Å ). Given a subset E of L

(ä) we write E

for the interior of E with respect to the functional jj Åjj

.

Theorem 1

Let fA (t) ; t 2 [0; T ] g be the one parameter family of nonlinear diãerential operators deåned in Deånition 1. Then we have

(a) A (t) is quasi-dissipative with respect to jj Åjj

, and has the resolvent J (ï ; t) = (I Ä ï A (t))

mapping X

into X

, for r > 0 and ï 2 (0; ã

), where ã

is an appropriately chosen positive constant.

(b) There exists on evolution operator fU (t; s) ; 0 î s î t î T g on L

(ä) such that for each v 2 L

(ä)

U (t; s)v = L

(ä)- lim

ï#0

Q

i=1

J (ï ; s + iï )v:

Theorem 2

Let R > 0, s 2 [0; T ) and v 2 X

(s). Then

(7) U (t; s)v = L

(ä) - lim

h#0

U

(t; s)v and the convergence is uniformly for t 2 [s; T ]. Moreover,

(8) U (t; s)v = L

(ä) - lim

h#0

[(tÄ

Y

i=0

C

(s + ih)v and the convergence is uniform for t 2 [s; T ].

The evolution operator fU (t; s) ; 0 î s î t î T g on L

(ä) is understood to provide solutions to (CP) in a generalized sense. In this talk we employ two notions of generalized solution to (CP).

Deånition 2

Let v 2 L

(ä). A function u 2 L

(ä Ç (0; T )) is called a weak solution of the problem (CP), if it satisåes the following two conditions:

(1) As an L

(ä)-valued function t 2 [0; T ] 7! u( Å ; t), u belongs to C([0; T ]; L

(ä)) and u( Å ; 0) = v;

(2) r å (u) 2 L

(0; T ; L

(ä)

) and for each ' 2 C

(ä Ç (0; T )), Z

0

Z

ä

(u'

+ A(x; t; u) År ' Ä B(x; t; u)' Ä r å (u) År ')dxdt = 0.

If in addition u 2 BV (ä Ç (0; T )), then u is said to be a BV -solution to the problem (CP).

Theorem 3

Let R > 0 and v 2 X

(0). Let u be a function on ä Ç [0; T ] deåned by u(x; t) = [ U (t; 0)v](x):

Then

(a) u is a distributional solution of (CP ) with initial value v.

(b) Assume lim sup

ú

ö (ú ; R) < 1 . Here ö (ú ; r) is modulus of time dependence. If v 2 X

\ BV (ä) satisåes lim inf

ï

jjJ (ï ; t)v Ä v jj

< 1 , then u 2 BV (ä), namely, u is a BV-solution to (CP ).

References

[1] L. C. Evans and R. Gariepy , Measure theory and åne properties of functions , Studies in Advanced Math., CRC Press, London,(1992).

[2] K. Kobayasi and S. Oharu, Japan J. Math., 10(1984), 243-270.

[3] K. Okamoto and S. Oharu, Adv, Math, Sci, Appl,. 8(1998), 581-629.

[4] W. P. Ziemer, Weakly diãerentiable functions, Springer-Verlag, New York, (1989).