Well-posedness of a needle crystal growth problem with anisotropic surface tension

2013 ◽  
Vol 93 (4) ◽  
pp. 698-728
Author(s):  
Xuming Xie
Keyword(s):  
Surface Tension ◽  
Crystal Growth ◽  
Well Posedness ◽  
Anisotropic Surface ◽  
Needle Crystal ◽  
Growth Problem

scholarly journals Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Classical Solutions ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation ◽  
Sivashinsky Equation

AbstractThe Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

scholarly journals Effect of anisotropic surface tension on deep cellular crystal growth in directional solidification

2014 ◽  
Vol 63 (3) ◽  
pp. 038101
Author(s):  
Chen Ming-Wen ◽  
Chen Yi-Chen ◽  
Zhang Wen-Long ◽  
Liu Xiu-Min ◽  
Wang Zi-Dong
Keyword(s):  
Surface Tension ◽  
Crystal Growth ◽  
Directional Solidification ◽  
Anisotropic Surface

EFFECT OF ANISOTROPIC SURFACE TENSION ON THE MORPHOLOGICAL STABILITY OF DEEP CELLULAR CRYSTAL GROWTH IN DIRECTIONAL SOLIDIFICATION

2019 ◽  
Vol 26 (06) ◽  
pp. 1850210
Author(s):  
HAN JIANG ◽  
MING-WEN CHEN ◽  
ZI-DONG WANG
Keyword(s):  
Surface Tension ◽  
Crystal Growth ◽  
Directional Solidification ◽  
Low Frequency ◽  
Expansion Method ◽  
Frequency Instability ◽  
Morphological Stability ◽  
Global Instability ◽  
Anisotropic Surface ◽  
Low Frequency Instability

This paper studies the effect of anisotropic surface tension on the morphological stability of deep cellular crystal in directional solidification by using the matched asymptotic expansion method and multiple variable expansion method. We find that the morphological stability of deep cellular crystal growth with anisotropic surface tension shows the same mechanism as that with isotropic surface tension. The deep cellular crystal growth contains two types of global instability mechanisms: the global oscillatory instability, whose neutral modes yield strong oscillatory dendritic structures, and the low-frequency instability, whose neutral modes yield weakly oscillatory cellular structures. Anisotropic surface tension has the significant effect on the two global instability mechanisms. As the anisotropic surface tension increases, the unstable domain of global oscillatory instability decreases, whereas the unstable domain of the global low-frequency instability increases.

scholarly journals $$H^1$$ solutions for a Kuramoto–Sinelshchikov–Cahn–Hilliard type equation

2021 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Cauchy Problem ◽  
Type Equation ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

AbstractThe Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

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.

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