The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered well-posedness results are proved.

A model describing the evolution of a binary mixture compressible, viscous, and macroscopically immiscible fluids is investigated. The existence global-in-time weak solutions for resulting system coupling compressible Navier–Stokes equations governing motion with Allen–Cahn equation order parameter proved without any restriction on size initial data.

We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng. 30 726–54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, continuum thermodynamically consistent is introduced. The resulting PDE system couples four different types of equations: Cahn–Hilliard type equation cells (which include proliferating and dead cells), Darcy law...

Galenko et al. proposed a modified Cahn–Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This contains an inertial term which causes the loss of any regularizing effect on solutions. Here we consider initial and boundary value problem for this two-dimensional bounded domain. We prove number results related well-posedness large time behavior In particular, analyze existence absorbing sets two different spaces and, correspondingly,...

We consider a model describing the evolution of tumor inside host tissue in terms parameters $\varphi_p$, $\varphi_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) $n$ (nutrient concentration). The variables satisfy Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved time), whereas obeys form Darcy law satisfies quasistatic diffusion equation. main novelty present work stands fact we able to configuration potential...

We consider a nonlinear parabolic system which governs the evolution of (relative) temperature \vartheta and an order parameter \chi . This describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling is characterized by singular potential W forces to take values in interval [-1,1] provide reasonable conditions on ensure that, from certain time on, stays uniformly away pure phases 1 -1 Combining this separation property with Łojasiewicz–Simon...

In the present work, we address a class of Cahn--Hilliard equations characterized by nonlinear diffusive dynamics and possibly containing an additional sixth order term. This model describes separation properties oil-water mixtures when substance enforcing mixing phases (a surfactant) is added. However, also closely connected with other Cahn--Hilliard-like relevant in different types applications. We first discuss existence weak solution to case configuration potential system has singular...

We study in this paper the well-posedness and asymptotic behavior, terms of global attractors, Caginalp system with coupled dynamic boundary conditions possibly singular potentials (e.g., logarithmic type).

A model describing the evolution of a liquid crystal substance in nematic phase is investigated terms three basic state variables: absolute temperature ϑ, velocity field u and director d, representing preferred orientation molecules neighbourhood any point reference domain. The time governed by incompressible Navier–Stokes system, with non-isotropic stress tensor depending on gradients where transport (viscosity) coefficients vary temperature. dynamics d described means parabolic equation...

We discuss a 3D model describing the time evolution of nematic liquid crystals in framework Landau-de Gennes theory, where natural physical constraints are enforced by singular free energy bulk potential proposed J.M. Ball and A. Majumdar.The thermal effects present through component that accounts for intermolecular interactions.The is consistent with general principles thermodynamics mathematically tractable.We identify priori estimates associated system evolutionary partial differential...

This paper addresses a doubly nonlinear parabolic inclusion of the form <p align="center"> $\mathcal A (u_t)+\mathcal B (u)$ ∋ f. align="left" class="times"> Existence solution is proved under suitable monotonicity, coercivity, and structure assumptions on operators $ B$, which in particular are both supposed to be subdifferentials functionals $L^2(\Omega)$. Since <i> unbounded</i> included analysis, this theory partly extends Colli & Visintin's work [24]. Moreover, additional hypotheses...

We consider a model of liquid crystals, based on nonlinear hyperbolic system differential equations, that represents an inviscid version the proposed by Qian and Sheng. A new concept dissipative solution is proposed, for which global-in-time existence theorem shown. The solutions enjoy following properties: (i) they exist globally in time any finite energy initial data; (ii) enjoying certain smoothness are classical solutions; (iii) coincides with strong originating from same data as long...

Abstract We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., exhibiting low Reynolds number), especially permeate porous medium. The system of equations consists Brinkman equation for the volume averaged velocity field and convective Cahn–Hilliard with dynamic boundary conditions phase field, which location two fluids within domain. are incorporated to interaction wall container more precisely. In particular, they allow evolution contact angle between...