scholarly journals Global well-posedness for the two-dimensional Muskat problem with slope less than 1

2019 ◽  
Vol 12 (4) ◽  
pp. 997-1022 ◽  
Author(s):  
Stephen Cameron
Keyword(s):  
Well Posedness ◽  
Two Dimensional ◽  
Muskat Problem

scholarly journals Well-posedness of the Muskat problem in subcritical L p -Sobolev spaces

2021 ◽  
pp. 1-43
Author(s):  
H. ABELS ◽  
B.-V. MATIOC
Keyword(s):  
Porous Medium ◽  
Vertical Motion ◽  
Parabolic Type ◽  
Immiscible Fluids ◽  
Well Posedness ◽  
Two Dimensional ◽  
Critical Space ◽  
Smoothing Property ◽  
Muskat Problem ◽  
Homogeneous Porous Medium

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an L p -setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

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

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

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.

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

Well-Posedness of Strongly Dispersive Two-Dimensional Surface Wave Boussinesq Systems

2012 ◽  
Vol 44 (6) ◽  
pp. 4195-4221 ◽  
Author(s):  
Felipe Linares ◽  
Didier Pilod ◽  
Jean-Claude Saut
Keyword(s):  
Surface Wave ◽  
Well Posedness ◽  
Two Dimensional ◽  
Boussinesq Systems ◽  
Dimensional Surface
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