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200

we obtain

-T(w,q)n

-an

- A

wdt'+ h{Mv)dt'

J ti J ti

= d(

+ j [n

•

T(u, r)n

- d]('

(t')dt' (57)

+ f

^-F(p)(,(t')dt'

^^df.

We consider w and r as a solution of the problem (55)-(56). Inequality (45) implies

sup |w

(-,s)|

«(n

)+ sup |w(-,s)|

+Q(nii)

+ sup \r(-,t)\

+a(Qb)

tl<S<to tl<S<to ti<S<t

••(

< c sup |f|c-(n

) + sup \h\

)

\ti+X<s<t

ti+A<s«o

sup

~^T + l

lc'+-.<'+-)/*(r,,x(*

,t„)) + A-(

1+a

sup |b(y,s)|

+x<

c,(

) r

x(ti+A,to)

+A"

sup |d(-,s)|c«(r

) + A-

sup |u(-,s)|c-(n

)

h+X<s<t

ti+\<s<t

+A M sup |u(-,s)|

i+a

,,)+ sup \r(;s)\

-(T

)

\*i+A<s<t

) ti+A<s<4

(58)

To estimate the norms of r and u in the right hand side, we use the boundary conditions

and the interpolation inequalities. We have

sup \b{y,s)\ < c sup |Vu(2/,s)|,

x((i+A,fo) Six(ti+\*o) (59)

(-,

)\c°(r

) < c(\u(;s)\

i+«

(rb)

+ \d\c°(r

) + \p{;s)\e'+<'(s

))

and

|u(-,s)|

i+a

(ni)

< e|u(-,s)|

2+»(n

) + ce-

5/2

||u(-,s)||

L2(n6)

\p(-,s)y+-(

) < £\p{-,s)\c3+°(s

) + ce~~

5/2

\\p(-,s)\\w}(s

)

Finally, when we consider (39)4 as

elliptic equation for p on Si and make use of

Schauder's estimates, we obtain

\p{-, s)|

«+«(s

) < c [\d + r - un

•

S(w)n

)(-, s)|

i+a

(Sl)

\\p(-,

s)\\

(Sl)

j .

Hence, the right hand side of (59)2 does not exceed

ce(\d(-,s)\

cl+a{rb)

+ \r{-,s)\

{Tb)

+ |w(-,s)|

2+«(n,,)) + \d\c°(r

)

+c(e)(l|w||

) + ||p(-,s

)ikc*>)-

Now, when we choose e = As' and taking account of (47), we obtain

201

sup |u

(-,s)|

,,)+ sup |u(-,s)|

2+a

(nt)

ti+2A<s<t

ti+2X<s<t

+ sup |r(-,s)|

i+«(n

)+ sup |p(-,s)|

(Sl)

ti+2A<s<(

ti+2A<s<t

/ sup |u(-,s)|

c0+

(ni

)

+ sup \r{-,s)y+^n

) \ ,

fin

(e'+

6) [

h+x<s<t

° ti+A<

<*o 160)

I + SUP |p(-,S)|

3+a

(Sl)

\ tl+A<s<t

+c(

')A~

7/2

( sup ||v(-,

)|U

2(sit)

+ sup

IIT/MH^SA

Vi<S<«0 «l<S<t

We define the function of

/(A) = A

7/2+a

( sup |u

(-,s)|

«(n

)+ sup |u(-,s)|

(ni>)

Vi+2A<s<!

ti+2X<s<t

+ sup |r(-,s)|

i+a

(f!b

)+ sup |p(-,s)|

3+o.(

;

ti+2A<s<to ti+2X<s<t

and we multiply (60) by

2+a

Since

T/2,

(60) implies

f(X)<c'2

(e' + S)f(\/2) + K,

where

K = c{e')[ sup ||u||

)+ sup \\p{-,s)\\

{Sl)

i ,

\tl<S<t

tl<S<t

is independent of A. We assume that c'2

2+a

6 < 1/4, and, choosing e' such that

c'e'2

2+Q

< 1/4, we obtain

which furnishes (51), once we take

T/2.

The proposition is proved. •

The following Corollary is a consequence of (33) and (55).

Corollary 7.5 For any arbitrary solution (u,r,p) of the problem (39) with the data sat-

isfying

+<»(ft

) + |Po|c

+«(Si) < £

and the condition

/ xx wocte = / / xx v

dSi,

Jfio JSiJ.R

which is defined in time interval (0,T), T < oo and satisfies (53) there holds the uniform

estimate

swpY

{u,r,p) < ce~

(|u

+«(ni,) + |po|c

+=(Si)) .

with the constants c and b independent of T.

202

Repeated application of the local existence theorem (Theorem 7.2) leads to our main

result Theorem 1.1. •

Acknowledgment

The authors are grateful to professor Taddia for his valuable comments. The au-

thors thank the GNFM-INDAM group for financial support. Padula thanks also the 60%

contract MURST.

References

[1] R.A. Brown and L.E. Scriven, Proc. R. Soc. London, A 371 (1980).

[2] F. Brulois, Var. Meth. Free

Surf.

Int. P. Concus and R. Finn Eds. Springer-Verlag,

Berlin.

[3] S. Chandrasekhar, Proc. R. Soc. London, A 286 (1965).

[4] I.Sh. Mogilevskii & V.A. Solonnikov, On the solvability of an evolution free boundary

problem for the Navier-Stokes equations in Holder spaces of functions, Mathematical

problems related to the Navier-Stokes equations, 11 (1992),

105-181.

[5] M. Padula, On the exponential decay to the rest state for a viscous isothermal fluid,

J. Fluid Mech. and Anal., 1 (1998).

[6] M. Padula & V.A. Solonnikov, On the Rayleigh-Taylor Stability, Annali dell'Univer-

sita' di Ferrara, (sez.VIII, Sci. Mat.) 46 (2000), 307-336.

[7] M. Padula & V.A. Solonnikov, On the global existence of nonsteady motions of a fluid

drop and their exponential decay to a uniform rigid rotation Quaderni di Matematica,

to appear.

[8] V.A. Solonnikov, Solvability of the problem of evolution of an isolated amount of

viscous incompressible capillary liquid, Zapiski Nauchn. Semin. LOMI, 140 (1984),

179-186.

[9] V.A. Solonnikov, Unsteady motion of a finite mass of fluid bounded by a free surface,

Zapiski Nauchnykh Sem. LOMI, 152 (1986), 137-157.

[10] V.A. Solonnikov, On an evolution of a isolated volume of viscous incompressible

capillary liquid for large values of time, Vestnik Leningrad Univ., ser.l, 3 (1987),

49-55.

[11] V.A. Solonnikov, On non-steady motion of a finite isolated mass of self-gravitating

fluid, Algebra and Analysis, 1 (1989), 207-249.

[12] V.A. Solonnikov, On the justification of the quasistationary approximation in the

problem of motion of a viscous capillary drop, Interfaces and free boundaries, 1

(1999),

125-173.

[13] A. V. Solormikov, Evolution free boundary value problems, Summer course in Punchal,

2000.

To appear.

On a convection-diffusion equation with partial

diffusivity *

Andrea Pascucci

Dipartimento di Matematica, Universita di Bologna ^

Abstract

We consider the Cauchy problem for the nonlinear degenerate equation in R

div(AVu) + u(b

•

Vti) - d

u =

f(-,u),

where A > 0 is a constant symmetric matrix and ker(A) is generated by b. We prove

the existence of a local viscosity solution u and we study the interior regularity of

u in the framework of Hormander type operators.

1 Introduction

We consider the Cauchy problem for the nonlinear convection-diffusion equation

div(AVu) + u(b

•

Vu)-d

u =

f{-,u),

in S

= R

x]0,T[, (1.1)

with initial datum

u{-,0)=g, inR*. (1.2)

We assume that /, g are globally Lipschitz continuous functions. Moreover we assume that

the matrix A is constant, symmetric and positive semidefmite. The convection direction

b is constant and generates ker(A).

Equations of form (1.1) were studied by Escobedo, Vazquez and Zuazua [11] in order to

describe the asymptotic behaviour as t

—)•

oo of solutions to a related parabolic equation

with complete diffusion. We remark that, without loss of generality, by performing a

suitable change of variables we may assume that A is diagonal so that b points along a

coordinate axis, for instance b = e

. Then it is convenient to denote a point in

(x, y) with x = (xi,..., x^-i) and y € R. Hence (1.1) becomes

Lu = A

u + ud

u-d

u =

f(-,u),

in ST, (1.3)

where A

denotes the Laplace operator acting in the x variables.

'Keywords: nonlinear degenerate parabolic equation, interior regularity, Hormander operators.

^Piazza di Porta

Donato 5, 40127 Bologna (Italy). E-mail: pascucci@dm.unibo.it

'Investigation supported by the University of

Bologna.

Funds for selected research topics.

204

205

Equation (1.3) also arises in mathematical finance, when studying agents' decisions

under risk. The classical approach for this financial problem is based on the representation

of agents' preferences in the framework of the utility theory and various models have been

proposed, aiming to taking into account many aspects of the dynamics of the economy.

Epstein and Zin in [10] propose a utility functional which is the solution of a backward

stochastic differential equation. Recently Antonelli, Barucci and Mancino [1] propose a

more sophisticated utility functional that considers some other aspects of decision making,

such as the agents' habit formation, which is described as a smoothed average of past

consumption and expected utility. In that model the couple of processes utility and habit

is described by a system of backward-forward stochastic differential equations. In [1] is

proved that there exists a unique solution u of such system, that satisfies some suitable

initial and final conditions, and which is a viscosity solution, in the sense of the User's

guide [9] of Cauchy problem (1.3)-(1.2). Moreover, in [1] it is proved that the solution u

is defined in a suitably small interval of time

[0,

T[ and satisfies

\u(x,y,t)-u{x',y',t)\ < c

(\x - x'\ + \y - y'\),

\u(x,y,t)-u{x,y,t')\<c

{l + \(x,y)\)\t-t'\^

for every (x, y), (x

, y') e R", t, t' €

[0,

T],

where

is a positive constant that depends on

the Lipschitz constants of / and g.

Related problems also arise in stochastic control theory. For instance, the value func-

tion v of a suitable control problem is a semiconcave solution of the following Cauchy

problem

v + ^(d

-d

v =

ip,

inR

x]0,T[,

v{-,0) =

il>,

inR

for some continuous functions

and ip (cf. [12]). Note that the function u — d

v is, at

least formally, a solution of a Cauchy problem like (1.3)-(1.2), even if it is only a locally

bounded function.

In this paper we are interested in the existence and interior regularity of local solutions

to the Cauchy problem (1.3)-(1.2). Our main results are stated in the next section.

Acknowledgments. The existence Theorem 2.1 is proved independently in a joint work

with Antonelli [2] and in [22] in collaboration with Polidoro. The interior regularity

problem is studied in [7], [8] with Citti and Polidoro in dimension three, and in [21] in

the general case.

2 Main results

Our aim is to find a functional space where this problem is well posed. The main difficulty

is the mixed parabolic-hyperbolic feature of equation (1.3) due to the lack of diffusion in

the y-direction, so that it may include the Burgers' equation, when / = 0 and g =

g{y).

We explicitly note that the nonlinearity in (1.3) is not monotone, then a standard

comparison principle does not hold and, as a consequence, the uniqueness of the solution

206

is not guaranteed. This fact also affects the existence of the solution. Indeed, when using

the classical Bernstein's method, a maximum principle for the operator Lv + v

(that

occurs when we differentiate both sides of (1.3) w.r.t. y) is required. Yet also more

sophisticated versions of that method (cf. Barles [3]) do not seem to work in our setting.

On the other hand, in the space of functions characterized by conditions (1.4) the operator

L in (1.3) does satisfy a comparison principle. Then we are able to prove the existence of

strong solution of the Cauchy problemfor small times. More precisely, we have

Theorem 2.1 Let f, g

globally

Lipschitz continuous. IfT>0is suitably small, then there

exists a unique function u, verifying estimates (1.4) on ST = K" x [0,T] and assuming

the initial datum g, such that

u e HiJSr), A

U e Ll

and equation (1.3) is satisfied a.e.

Let us remark that, in general, the linear growth of the initial datum g does not allow

solutions which are defined globally in t. Indeed, let us consider the following simple

example: for N = 2, take / = 0 and g(x, y) = x + y. A direct computation shows that

u(x,y,t) = jz^ is the unique solution to (1.3)-(1.2) and it blows up as t

—¥

Our main results regard the interior regularity of the strong solution u of Theorem 2.1.

Since L is a degenerate second order operator, the known results by Cabre e Caffarelli [5],

Trudinger [24], Bian e Dong [4], Wang [25] do not apply. Therefore we set the problem

in the framework of subelliptic operators on nilpotent Lie groups. We remark that L

is an operator such that the matrix of the coefficients of the second order derivatives is

only positive semi-definite. As one can expect, the solution u of the equation Lu = 0 is

smooth in the directions in which the matrix is non-degenerate, but not in other directions.

Consider for example the operator

= dl

(2.1)

in the variables (x,y,t) e R

. Every solution u of L\u = 0 is smooth with respect to the

variables x and y, but is not regular in the variable t. However, as Hormander pointed

out in the celebrated paper [16], there are other "regularity directions" for the solution

u, and these directions are the ones of the commutators. For instance, let us consider the

Kohn-Laplace operator in R

= (d

+ 2yd

f + {d

- 2xd

f . (2.2)

As before, there are only two directional derivatives, while the dimension of the space is

three, but in this case every solution of L

u = 0 is smooth, not only in the directions

of the derivatives X = d

2yd

and Y = d

—

2xd

but also in the direction of their

commutator

[X,Y} = XY -YX = -4d

(

The operator considered above is a simple but meaningful example of the class studied

by Hormander in [16]. Let Xo,

...,X

be a set of linear first order operators (i.e. vector

fields) defined as

i=l

207

where

are smooth functions on some domain SlcK" and let also

G C°°(fi). Horman-

der proved

[16] that

if u is a

solution of the equation

^XiXjU +

u = f, in Q, (2.3)

and the Lie algebra generated by the vector fields

, ...,X

has rank

n at

every point of

Q, then

u €

C°°(U).

Hormander's result was the starting point of an extensive research aiming to investigate

the regularity properties

the operators

(2.3) and their links with some suitable Lie

group structures on R". The existence of a fundamental solution and of a control distance

have been established

[20], [23], [17]. Using these properties,

general theory

the

regularity both

Sobolev spaces and

spaces

Holder continuous functions has been

settled down

[13], [14], [23] and [18].

Aiming

use the linear theory

for

the study

our problem we can

try to

consider

the "linearized" operator

= A

+ ud

- d

where

u is

considered

as a

coefficient, but we immediately realize that the smoothness of

the coefficients

is a fundamental assumption in the previous papers and, in our problem,

we cannot assume that

the

coefficient

u of

the equation

C°°, since

the

smoothness

the solution

u is

exactly the goal

our study.

Actually, in all the papers cited above

is crucial that the vector fields

are regular

at least

as it is

sufficient

obtain, by commutation,

linearly independent vector fields

at every point of R". For the first time, Franchi and Lanconelli [15] studied the properties

control distance related

to a

family

non regular vector fields, aiming

adapt

the

classical Moser's iteration scheme

prove the Holder regularity

weak solutions

equation of the form

where the matrix (aij(x))

positive semi-definite

for

every i6l". Franchi

Lanconelli

assume that

the

operator

"elliptic" w.r.t.

family

X\, ...,X

Lipschitz continuous

vector fields,

the sense that

3=1

i=i

for some positive constant A. This class of operators includes,

for

instance,

in R

\x\

dl,

where

a is a

positive constant. We stress that

general theory

for

operators with non-

smooth coefficients

not available.

Here we employ

technique introduced by Citti

[6] where the author considers the

regularity of the solutions

the following equation

prescribed Levi curvature

- _L ,

+ u

l , «

y -

+ u

+ u} +

)

Cu =

Y^uu

(

2^^3-u*

= k ^ ,

208

in the variables (x, y, ()eE

[6]

is pointed out that the principal part of the above

operator can be written

the form (2.3)

Cu =

+ Y

in terms of the nonlinear vector fields

= d

-UxU

= d

U_x_^U

idt

+ uf

Then, based on the notion

"intrinsic" Taylor expansion

the coefficients

the op-

erator,

modification

the freezing method used

Rothschild and Stein

[23]

developed.

Analogously, we remark that, by letting

Xj = d

j =

1,...,

and X

= ud

- d

then

in (1.3) can be formally represented

= E

°-

(

)

Since the commutator of

and

[Xj,X

]

u)d

the Hormander condition is satisfied if

u(x,y,t)^0,

(2.5)

for every

(x,y,t),

for some index

Note that the regularity of the solution

in Theorem

2.1

stated

condition (1.4), then d

defined almost everywhere and the above

condition has

be considered only formally. Then we first state

regularity result

for

classical solutions to (1.3).

Theorem 2.2 Let Q be

open

set in

N+1

and

u a

classical solution

(1.3) on

with

f e

C°°.

(2.5) holds, then

u e

C°°(fi).

If we

do not

require any assumption

on the

commutators

(in

particular,

not require anymore condition (2.5)) then the Lie algebra associated

the operator

completely unknown. However we consider

L as a

subelliptic operator with respect

some tentative Lie groups. This allows us to prove the existence of the derivatives d

and

XQU,

defined as the directional derivative with respect to the vector

(0,

u(z), — 1)

at the point

= (x, y,

u(z)

= —(z) =

hm -^

^ ^-. 2.6

Then we have

209

Theorem 2.3 The strong solution u to (1.3) in Theorem 2.1 is a classical solution in

the sense that d

.u, j = 1,..., N

—

1, and

XQU

are continuous functions and the equation

is satisfied at every point.

This result is quite reasonable, since without assuming the Hormander condition we are

able to prove the regularity of u only in the directions of the vector fields. Note that,

although the derivative J^- can be obtained as a sum of more simple terms

udyU

—

dfU

it is not true in general that the terms ud

u and d

u are continuous. Also note that, at this

point, the Hormander condition (2.5) is meaningful since XjU are defined and continuous

functions.

3 Existence

The proof of Theorem 2.1 is based on some estimates which can be obtained by adapting

the classical Bernstein's method. We consider the regularized Cauchy-Dirichlet problem

in a cylinder

Llu = A

u + e

u + vd

u-d

u =

f(;v),

in Q

= Bx]0,T[, (3.1)

u = g, in d

- (3.2)

where e > 0, B is a ball in R

and

denotes the "parabolic" boundary of QT

defined as (B x {0}) U (dB x [0,T]). By a standard density argument we may assume

that /,j£ C°°nLip. We fix a positive constant Ci such that, for (x, y, t, v) e Q

, it holds

c\ > max{Lipschitz constants of /, g},

l<?(z,y)l<ciVi

l(z-y)l

. \f(x,y,t,v)\<

^i + \(x,

,t,v)\i.

Given a €]0,1[, we assume that the coefficient v in (3.1)-(3.2) belongs to Ci

), the

Holder space w.r.t. to the parabolic distance d((x,y,t), (x',y',t')) = \x

—

x'\ + \y

—

y'\ +

—

t'|5. Moreover we assume that v satisfies in QT the estimates

\v(x,y,t)\<2

^l +

\(x,y)\\

(3.4)

v\<2c

i =

l,...,N-l,

|a„w| < 2ci. (3.5)

The following proposition is the key step in the proof of Theorem 2.1 which then follows

by the Schauder's fixed point theorem and passing at limit as e goes to zero.

Proposition 3.1 There exists T > 0 such that, under the above assumptions, every

classical solution of (3.1)-(3.2) verifies the e-uniform estimates (3.4), (3.5).

Proof.

Let a be a classical solution of (3.1)-(3.2). We prove estimate (3.4) for u by

applying the maximum principle to the functions H ±u where H is defined as follows

H{x, y, t) = (

+ fit) ^l + \{x,yW