scholarly journals On the Parabolicity of the Muskat Problem: Well-Posedness, Fingering, and Stability Results

2011 ◽  
pp. 193-218 ◽  
Author(s):  
Joachim Escher ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Well Posedness ◽  
Muskat Problem ◽  
Stability Results

scholarly journals Well-posedness and stability results for a quasilinear periodic Muskat problem

2019 ◽  
Vol 266 (9) ◽  
pp. 5500-5531 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Well Posedness ◽  
Muskat Problem ◽  
Stability Results

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.

scholarly journals A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

2019 ◽  
Vol 62 (3) ◽  
pp. 417-444
Author(s):  
A. Chambolle ◽  
M. Holler ◽  
T. Pock
Keyword(s):  
Image Reconstruction ◽  
Inverse Problems ◽  
Incomplete Data ◽  
Variational Model ◽  
Well Posedness ◽  
Optimal Solutions ◽  
Analytical Properties ◽  
Numerical Performance ◽  
Stability Results ◽  
Continuous Setting

AbstractA variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is convex and allows for simultaneous image reconstruction and atom learning in a general, inverse problems context. Further, motivated by an improved numerical performance, also a semi-convex variant is included in the analysis and the experiments of the paper. For both settings, fundamental analytical properties allowing in particular to ensure well-posedness and stability results for inverse problems are proven in a continuous setting. Exploiting convexity, globally optimal solutions are further computed numerically for applications with incomplete, noisy and blurry data and numerical results are shown.

scholarly journals On well-posedness of the Muskat problem with surface tension

2020 ◽  
Vol 374 ◽  
pp. 107344
Author(s):  
Huy Q. Nguyen
Keyword(s):  
Surface Tension ◽  
Well Posedness ◽  
Muskat Problem

scholarly journals Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity

2018 ◽  
Vol 99 (1) ◽  
pp. 50-74 ◽  
Author(s):  
Mohammad M. Al-Gharabli ◽  
Aissa Guesmia ◽  
Salim A. Messaoudi
Keyword(s):  
Asymptotic Stability ◽  
Viscoelastic Plate ◽  
Well Posedness ◽  
Plate Equation ◽  
Stability Results

scholarly journals Well-posedness and stability results for nonlinear abstract evolution equations with time delays

2018 ◽  
Vol 18 (2) ◽  
pp. 947-971 ◽  
Author(s):  
Serge Nicaise ◽  
Cristina Pignotti
Keyword(s):  
Time Delays ◽  
Evolution Equations ◽  
Well Posedness ◽  
Abstract Evolution Equations ◽  
Stability Results

scholarly journals Well-posedness and stability results for the Gardner equation

2011 ◽  
Vol 19 (4) ◽  
pp. 503-520 ◽  
Author(s):  
Miguel A. Alejo
Keyword(s):  
Well Posedness ◽  
Gardner Equation ◽  
Stability Results
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