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In 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.

The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces