AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$
W
p
s
(
R
)
, where $${p\in (1,2]}$$
p
∈
(
1
,
2
]
and $${s\in (1+1/p,2)}$$
s
∈
(
1
+
1
/
p
,
2
)
. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$
W
p
s
¯
-
2
(
R
)
, where $${\overline{s}\in (1+1/p,s)}$$
s
¯
∈
(
1
+
1
/
p
,
s
)
. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.
Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system