236
4. On the free boundary for Budyko type models.
This section is devoted to present some qualitative properties of the solutions associated to the Budyko type model. The discontinuity of the albedo function assumed in the
Budyko model generates a natural free boundary or interface ((t) between the ice-covered
area ({x E I: u(x,t) < -10}) and the ice-free area ({x E I: u(x,t) > -10}). The free
boundary is then given as ((t) = {x E I: u(x, t) = -10}. In Xu [1991] the Budyko model
for p = 2 is considered. He shows that if the initial datum Uo satisfies
uo(x) = uo( -x), Uo E C 3 ([-1, 1]), u~(x) < 0 for any x E (0,1)
and there exists ((0) E (0,1) such that (uo(x) + 10)(x - ((0)) < 0
for any x E [0, ((0)) U (((0), 1],
then there exists a bounded weak solution u of (P) for which the set ((t) = {(+(t)} U
{(-(t)} with x = (+(t) a smooth curve, (_(t) = (+(t) and (+0 E COO([O, T*)) where T*
is characterized as the first time t for which (+(t) = 1. He also gives an expression for the
derivative (~(t) (some related results for a model corresonding to p(x) = 1 can be found
in Feireisl-Norbury [1991]). We point out that the uniqueness result (Theorem 4) can be
applied for such an initial datum (see Remark 4).
The size of the separating zone ((t) for other models is in fact a controversial question.
So, some satellite pictures (Image of the Weddell sea taken by the satellite Spot on
December 10, 1987: Lions [1991]) show that the separating region between the ice-free
and the ice-covered zones is not a simple line on the Earth (i.e. a point in (-1,0) or
(0,1)) but a narrow zone where ice and water are mixed. Mathematically it corresponds
to say that the set
M(t) = {x E I: u(x,t) = -10}
is a positively measured set. In the following we shall denote this set as the mushy region
(since it plays the same role than in changing phase problems, see e.g. Dfaz-FasanoMeirmanov [1992]).
Using the strong maximum principle (see e.g. Vazquez [1984]) it is possible to show
that if p = 2 (or more in general if 1 < p:::; 2) the interior set of the mushy region M(t)
is empty even if the interior of M(O) is a nonempty open set. The main goal of the next
result is to show that this is not the case when p > 2 (as it happens for the Stone model
o
: p = 3). A necessary condition for M(t):f 0 is that Ra(x, t, -10) - Re(x, t, -10) '30 for
any x EM (t) and t E [0, T]. In the case of the Budyko model Ra is defined by (4) and
(5), Re by (7) and the necessary condition can be written in the following terms
A-lOBE [aiQ(x,t),ajQ(x,t)]fora.e. xEI, a.e. tE[O,T]
(44)
We shall show that if p > 2 this condition is also sufficient.
4. On the free boundary for Budyko type models.
This section is devoted to present some qualitative properties of the solutions associated to the Budyko type model. The discontinuity of the albedo function assumed in the
Budyko model generates a natural free boundary or interface ((t) between the ice-covered
area ({x E I: u(x,t) < -10}) and the ice-free area ({x E I: u(x,t) > -10}). The free
boundary is then given as ((t) = {x E I: u(x, t) = -10}. In Xu [1991] the Budyko model
for p = 2 is considered. He shows that if the initial datum Uo satisfies
uo(x) = uo( -x), Uo E C 3 ([-1, 1]), u~(x) < 0 for any x E (0,1)
and there exists ((0) E (0,1) such that (uo(x) + 10)(x - ((0)) < 0
for any x E [0, ((0)) U (((0), 1],
then there exists a bounded weak solution u of (P) for which the set ((t) = {(+(t)} U
{(-(t)} with x = (+(t) a smooth curve, (_(t) = (+(t) and (+0 E COO([O, T*)) where T*
is characterized as the first time t for which (+(t) = 1. He also gives an expression for the
derivative (~(t) (some related results for a model corresonding to p(x) = 1 can be found
in Feireisl-Norbury [1991]). We point out that the uniqueness result (Theorem 4) can be
applied for such an initial datum (see Remark 4).
The size of the separating zone ((t) for other models is in fact a controversial question.
So, some satellite pictures (Image of the Weddell sea taken by the satellite Spot on
December 10, 1987: Lions [1991]) show that the separating region between the ice-free
and the ice-covered zones is not a simple line on the Earth (i.e. a point in (-1,0) or
(0,1)) but a narrow zone where ice and water are mixed. Mathematically it corresponds
to say that the set
M(t) = {x E I: u(x,t) = -10}
is a positively measured set. In the following we shall denote this set as the mushy region
(since it plays the same role than in changing phase problems, see e.g. Dfaz-FasanoMeirmanov [1992]).
Using the strong maximum principle (see e.g. Vazquez [1984]) it is possible to show
that if p = 2 (or more in general if 1 < p:::; 2) the interior set of the mushy region M(t)
is empty even if the interior of M(O) is a nonempty open set. The main goal of the next
result is to show that this is not the case when p > 2 (as it happens for the Stone model
o
: p = 3). A necessary condition for M(t):f 0 is that Ra(x, t, -10) - Re(x, t, -10) '30 for
any x EM (t) and t E [0, T]. In the case of the Budyko model Ra is defined by (4) and
(5), Re by (7) and the necessary condition can be written in the following terms
A-lOBE [aiQ(x,t),ajQ(x,t)]fora.e. xEI, a.e. tE[O,T]
(44)
We shall show that if p > 2 this condition is also sufficient.