scholarly journals Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces

2021 ◽  
Vol 207 ◽  
pp. 112267
Author(s):  
Roberto de A. Capistrano-Filho ◽  
Milena Monique de S. Gomes
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Well Posedness ◽  
Kawahara Equation

scholarly journals SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

2010 ◽  
Vol 20 (01) ◽  
pp. 95-120 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
FERNANDO LÓPEZ GARCÍA
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Basic Result ◽  
Weighted Sobolev Spaces ◽  
Mean Value ◽  
Well Posedness ◽  
Existence Of A Solution ◽  
Simply Connected ◽  
Distance To The Boundary

If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution [Formula: see text] for some r < 2 depending on the power of the cusp.

The Kawahara equation in weighted Sobolev spaces

Nonlinearity ◽  
2008 ◽  
Vol 21 (7) ◽  
pp. 1489-1505 ◽  
Author(s):  
Netra Khanal ◽  
Jiahong Wu ◽  
Juan-Ming Yuan
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Kawahara Equation

Well-posedness in weighted Sobolev spaces for elliptic equations of Cordes type

2015 ◽  
Vol 67 (3) ◽  
pp. 539-554 ◽  
Author(s):  
Loredana Caso ◽  
Roberta D’Ambrosio ◽  
Maria Transirico
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Weighted Sobolev Spaces ◽  
Well Posedness

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 Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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