A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion

Kettani, Perla El; Hilhorst, Danielle; Lee, Kai

Displaying similar documents to “A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion”

Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift

Ross G. Pinsky (2014)

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

Similarity:

We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases – a transient to $+ \infty$ mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain non-negative constant when the diffusion is sufficiently...

Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík (2021)

Czechoslovak Mathematical Journal

Similarity:

We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space $ℝ^{d}$ driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted $L^{2}$ -space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.

Geometry of non-holonomic diffusion

Simon Hochgerner, Tudor S. Ratiu (2015)

Journal of the European Mathematical Society

Similarity:

We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For $G$ -Chaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard. ...

Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents

Aya Khaldi, Amar Ouaoua, Messaoud Maouni (2022)

Mathematica Bohemica

Similarity:

We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $u_{t} - M (\int_{Ω} {| \nabla u |}^{2} d x) Δ u + {| u |}^{m (x) - 2} u_{t} = {| u |}^{r (x) - 2} u .$ We prove with suitable assumptions on the variable exponents $r (\cdot),$ $m (\cdot)$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.

On the stability of the ALE space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

Monika Balázsová, Miloslav Feistauer (2015)

Applications of Mathematics

Similarity:

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary...

Euler's Approximations of Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau-Giłka (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let D be either a convex domain in $ℝ^{d}$ or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman (1984) and Saisho (1987). We investigate convergence in law as well as in $L^{p}$ for the Euler and Euler-Peano schemes for stochastic differential equations in D with normal reflection at the boundary. The coefficients are measurable, continuous almost everywhere with respect to the Lebesgue measure, and the diffusion coefficient may degenerate on some subsets of the domain. ...

Asymptotics for conservation laws involving Lévy diffusion generators

Piotr Biler, Grzegorz Karch, Wojbor A. Woyczyński (2001)

Studia Mathematica

Similarity:

Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their $L^{p}$ -decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

Remarks on balanced norm error estimates for systems of reaction-diffusion equations

Hans-Goerg Roos (2018)

Applications of Mathematics

Similarity:

Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^{1}$ seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.

Analysis of the discontinuous Galerkin finite element method applied to a scalar nonlinear convection-diffusion equation

Hozman, Jiří, Dolejší, Vít

Similarity:

We deal with a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the discontinuous Galerkin finite element method. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact...

On estimation of diffusion coefficient based on spatio-temporal FRAP images: An inverse ill-posed problem

Kaňa, Radek, Matonoha, Ctirad, Papáček, Štěpán, Soukup, Jindřich

Similarity:

We present the method for determination of phycobilisomes diffusivity (diffusion coefficient $D$ ) on thylakoid membrane from fluorescence recovery after photobleaching (FRAP) experiments. This was usually done by analytical models consisting mainly of a simple curve fitting procedure. However, analytical models need some unrealistic conditions to be supposed. Our method, based on finite difference approximation of the process governed by the Fickian diffusion equation and on the minimization...

Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed

Marek Fila, Michael Winkler (2008)

Journal of the European Mathematical Society

Similarity:

We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x = 0$ . We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x = 0$ while $u (x, T)$ remains bounded for $x > 0$ .

On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains

Weronika Łaukajtys (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let D be an open convex set in $ℝ^{d}$ and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: $X_{t} = H_{t} + \int_{0}^{t} ⟨ F {(X)}_{s -}, d Z_{s} ⟩ + K_{t}$ , t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.