Modelling of multicomponent diffusive phase transformation in solids

Vala, Jiří

Displaying similar documents to “Modelling of multicomponent diffusive phase transformation in solids”

Homogenization of a three-phase composites of double-porosity type

Ahmed Boughammoura, Yousra Braham (2021)

Czechoslovak Mathematical Journal

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In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size $ε^{β}$ ( $ε > 0$ and $β > 0$ ) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order $ε^{2}$ (the so-called double-porosity type scaling) while the matrix material has a conductivity...

Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality

Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2013)

Journal of the European Mathematical Society

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We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{M I X}$ is conjectured to be polynomial in $L$ . In [37] it was shown that for a large enough inverse-temperature $β$ and...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form $u_{t} = u_{x x} + f (u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f (u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f (u)$ which is $C^{1}$ and satisfies $f (0) = 0$ , we show that the omega limit set $ω (u)$ of every bounded positive solution is determined by a stationary...

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{- β}$ , with $β \in [0, \infty)$ . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$ . If $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...

Vectorial quasilinear diffusion equation with dynamic boundary condition

Nakayashiki, Ryota

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In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) $_{ε}$ with a nonnegative constant $ε$ , and for any $ε \geq 0$ , (S) $_{ε}$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $Ω$ , and the parabolic equation on the boundary $Γ : = \partial Ω$ , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

Nonlinear diffusion equations with perturbation terms on unbounded domains

Kurima, Shunsuke

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This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $u_{t} + (- Δ + 1) β (u) + G (u) = g in Ω \times (0, T)$ in an unbounded domain $Ω \subset ℝ^{N}$ with smooth bounded boundary, where $N \in ℕ$ , $T > 0$ , $β$ , is a single-valued maximal monotone function on $ℝ$ , e.g., $β (r) = {| r |}^{q - 1} r (q > 0, q \neq 1)$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${(H^{1} (Ω))}^{*}$ to ${(H^{1} (Ω))}^{*}$ . The present work establishes existence and estimates for the above problem.

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

Annales de l'I.H.P. Probabilités et statistiques

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A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K (t), i (t), Y (t))$ on $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , where $𝕋^{2}$ is the two-dimensional torus. Here $(K (t), i (t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y (t)$ is an additive functional of $K$ , defined as $\int_{0}^{t} v (K (s)) d s$ , where $| v | \sim 1$ for small $k$ . We prove that the rescaled process ${(N ln N)}^{- 1 / 2} Y (N t)$ converges in distribution to a two-dimensional Brownian motion. As a consequence,...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u ₜ - Δ (a_{11} u) = h (t, x) {| v |}^{p}$ , $v ₜ - Δ (a_{21} u) - Δ (a_{22} v) = k (t, x) {| w |}^{q}$ , $w ₜ - Δ (a_{31} u) - Δ (a_{32} v) - Δ (a_{33} w) = l (t, x) {| u |}^{r}$ , for $x \in ℝ^{N}$ , t > 0, p > 0, q > 0, r > 0, $a_{i j} = a_{i j} (t, x, u, v)$ , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x \in ℝ^{N}$ , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

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In a cholesteric liquid crystal the director field $n (x, y, z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

Numerical approximation of the non-linear fourth-order boundary-value problem

Svobodová, Ivona

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We consider functionals of a potential energy $ℙ_{ψ} (u)$ corresponding to $𝑎𝑛 𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 - 𝑣𝑎𝑙𝑢𝑒 𝑝𝑟𝑜𝑏𝑙𝑒𝑚$ . We are dealing with $𝑎 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑡ℎ𝑖𝑛 𝑎𝑛𝑛𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒$ with $𝑁𝑒𝑢𝑚𝑎𝑛𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠$ . Various types of the subsoil of the plate are described by various types of the $𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒$ nonlinear term $ψ (u)$ . The aim of the paper is to find a suitable computational algorithm.

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system $u_{i t} - d_{i} Δ u_{i} = \prod_{k = 1}^{m} u_{k}^{p_{k}^{i}}$ , i = 1,…,m, $x \in ℝ^{N}$ , t > 0, $u_{i} (0, x) = u_{0 i} (x)$ , i = 1,…,m, $x \in ℝ^{N}$ , where m > 1 and $p_{k}^{i} > 0$ , to describe asymptotics near the blow up point.

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D \subset ℝ^{d}$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$ . We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$ . Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$ , we provide results concerning the well-posedness...

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

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We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_{0} \in H^{s - 1 + ε}$ , $w_{0} \in H^{s - 1}$ and $b_{0} \in H^{s}$ for $s > \frac{3}{2}$ and any $0 < ε < 1$ . The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$ .