Abstract This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on potential, we prove strict both two three-space dimensions. Namely, 2D, notably extend minimal potential adopted so far literature, only requiring a mild growth condition of its first derivative near points $\pm 1$ , without any pointwise...

We consider a multi-component version of the conserved Allen–Cahn equation proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for phase separation. In our case, free energy is characterized mixing entropy density which belongs to large class physically relevant entropies like, example, Boltzmann–Gibbs entropy. establish well-posedness Cauchy–Neumann problem with respect natural notion (finite) solution more regular under appropriate assumptions strictly separated from...

We study the existence of weak solutions and corresponding sharp interface limit an anisotropic Cahn-Hilliard equation with disparate mobility, i.e., mobility is degenerate in one two pure phases, making diffusion that phase vanish. The double-well potential polynomial weighted by a spatially inhomogeneous coefficient. In when parameter width tends to zero, under energy convergence assumption, we prove converge BV Hele-Shaw flow. also add some numerical simulations analyze effects anisotropy...

Abstract We show global in time existence and uniqueness on any finite interval of strong solutions to a Navier–Stokes/Cahn-Hilliard type system given two-dimensional evolving surface the case different densities singular (logarithmic) potential. The describes diffuse interface model for two-phase flow viscous incompressible fluids an surface. also establish validity instantaneous strict separation property from pure phases. To these results we use our previous achievements local...

We investigate the nonlocal version of Abels-Garcke-Grün (AGG) system, which describes motion a mixture two viscous incompressible fluids. This consists Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, an additional flux term due to interface diffusion. In particular, Cahn-Hilliard dynamics concentration (phase-field) is governed aggregation/diffusion competition Helmholtz free energy with singular (logarithmic) potential constant mobility....

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with singular Flory-Huggins potential in three-dimensional setting. The enters governing state system a nonlinear fashion form of prescribed solenoidal, that is divergence-free, vector field, whereas cost functional to be minimized tracking-type. novelties present paper are twofold: addition application, intrinsic difficulties optimization forced us first establish new...

Abstract We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation an ideal mixture $$N\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> components occupying bounded region $$\Omega \subset \mathbb {R}^{d},$$ <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> $$d\le 3$$...

.We consider the Cahn–Hilliard equation with constant mobility and logarithmic potential on a two-dimensional evolving closed surface embedded in \(\mathbb{R}^3\), as well related weighted model. The well-posedness of weak solutions for corresponding initial value problems given time interval \([0,T]\) have already been established by first two authors. Here we prove some regularization properties finite time. Then show validity strict separation property both problems. This means that stay...

We consider the nonlocal Cahn-Hilliard equation with singular (logarithmic) potential and constant mobility in three-dimensional bounded domains we establish validity of instantaneous strict separation property. This means that any weak solution, which is not a pure phase initially, stays uniformly away from phases $\pm1$ positive time on. work extends result dimension two for same gives answer to long standing open problem property dimensions higher than two. In conclusion, show how this...

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the heat equation and subject to Dirichlet-type boundary conditions. viscosity is supposed depend on temperature stress depends strain through suitable power law depending [Formula: see text] (shear thinning case). For this problem we establish existence of weak solution as well prove some regularity results both Stokes cases. Then, latter case Carreau approximated FEM scheme error estimates...

We consider a phase-field model which describes the interactions between blood flow and thrombus. The latter is supposed to be viscoelastic material. potential describing cohesive energy of mixture assumed Flory-Huggins type (i.e. logarithmic). This ensures boundedness from below dissipation energy. In two dimensional case, we prove local (in time) existence uniqueness strong solution, provided that viscosities pure fluid phases are close enough. also show order parameter remains strictly...

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The consists of the Navier–Stokes–Voigt equations where instantaneous kinematic viscosity has been replaced by memory term incorporating hereditary effects coupled with Cahn–Hilliard equation Flory–Huggins potential. resulting system is subject to no‐slip condition (volume averaged) velocity and no‐flux boundary conditions order parameter chemical potential . first establish well‐posedness initial...

In this paper we address the importance and impact of employing structure preserving neural networks as surrogate analytical physics-based models typically employed to describe rheology non-Newtonian fluids in Stokes flows. particular, propose test on real-world scenarios a novel strategy build data-driven rheological based use Input-Output Convex Neural Networks (ICNNs), special class feedforward network scalar valued functions that are convex with respect their inputs. Moreover, show,...

The main goal of this paper is to establish the nonlocal-to-local convergence strong solutions a Navier--Stokes--Cahn--Hilliard model with singular potential describing immiscible, viscous two-phase flows matched densities, which referred as Model H. This means that we show nonlocal H converge solution local weight function in interaction kernel approaches delta distribution. Compared previous results literature, our novelty further corresponding rates. Before investigating convergence,...

We show existence and uniqueness of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on given two-dimensional evolving surface in the case different densities singular (logarithmic) potential. The describes diffuse interface model for two-phase flow viscous incompressible fluids an surface. also establish validity instantaneous strict separation property from pure phases. To these results we use our previous achievements local well-posedness together with suitable novel...

We focus on the derivation and analysis of a model for multi-component phase separation occurring biological membranes, inspired by observations lipid raft formation. The integrates local membrane composition with curvature, describing membrane's geometry through perturbation method represented as graph over an undeformed Helfrich minimising surface, such sphere. resulting energy consists small deformation functional coupled to Cahn-Hilliard functional. By applying Onsager's variational...

We investigate the initial-value problem for incompressible tangential Navier-Stokes equation with variable viscosity on a given two-dimensional surface without boundary. Existence of global weak and strong solutions under inhomogeneous forcing is proved by fixed-point continuation argument. Continuous dependence data, backward uniqueness, instantaneous regularization are also discussed. Depending effect dissipative nondissipative components system, we long-time behavior solutions. prove...