170 research outputs found

    Lattice Boltzmann Methods for Partial Differential Equations

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    Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions

    Beginner's guide to Aggregation-Diffusion Equations

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    The aim of this survey is to serve as an introduction to the different techniques available in the broad field of Aggregation-Diffusion Equations. We aim to provide historical context, key literature, and main ideas in the field. We start by discussing the modelling and famous particular cases: Heat equation, Fokker-Plank, Porous medium, Keller-Segel, Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez, McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present the well-posedness frameworks given as PDEs in Sobolev spaces, and gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time, for which we need to understand minimisers of a free energy. We then present some numerical methods which have been developed. We conclude the paper mentioning some related problems

    Global behavior of nonlocal in time reaction-diffusion equations

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    The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation ∂t(k∗(u−u0))+Lx[u]=f(u),    x∈Ω⊂Rn,t>0,\partial_t (k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\in\Omega\subset\mathbb{R}^n, t>0, where k∈Lloc1(R+),k\in L^1_{loc}(\mathbb{R}_+), ff is a locally Lipschitz function, Lx\mathcal{L}_x is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented.Comment: 16 page

    Decay estimates for the time-fractional evolution equations with time-dependent coefficients

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    In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator, and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source.Comment: 23 pages. The previous version of the paper has been edited according to the comments of the reviewer

    Depleting the signal: Analysis of chemotaxis-consumption models—A survey

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    We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

    Propagation and reaction–diffusion models with free boundaries

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    In this short survey, we describe some recent developments on the modeling of propagation by reaction-differential equations with free boundaries, which involve local as well as nonlocal diffusion. After the pioneering works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP) and Skellam, the use of reaction–diffusion equations to model propagation and spreading speed has been widely accepted, with remarkable progresses achieved in several directions, notably on propagation in heterogeneous media, models for interacting species including epidemic spreading, and propagation in shifting environment caused by climate change, to mention but a few. Such models involving a free boundary to represent the spreading front have been studied only recently, but fast progress has been made. Here, we will concentrate on these free boundary models, starting with those where spatial dispersal is represented by local diffusion. These include the Fisher–KPP model with free boundary and related problems, where both the one space dimension and high space dimension cases will be examined; they also include some two species population models with free boundaries, where we will show how the long-time dynamics of some competition models can be fully determined. We then consider the nonlocal Fisher–KPP model with free boundary, where the diffusion operator Δu is replaced by a nonlocal one involving a kernel function. We will show how a new phenomenon, known as accelerated spreading, can happen to such a model. After that, we will look at some epidemic models with nonlocal diffusion and free boundaries, and show how the long-time dynamics can be rather fully described. Some remarks and comments are made at the end of each section, where related problems and open questions will be briefly discussed

    Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

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    We consider conservative and gradient flows for NN-particle Riesz energies with mean-field scaling on the torus Td\mathbb{T}^d, for d≥1d\geq 1, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range d−2≤s<dd-2\leq s<d not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called W˙−1,∞\dot{W}^{-1,\infty} kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.Comment: 63 page

    A class of fractional parabolic reaction-diffusion systems with control of total mass: theory and numerics

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    In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of RN\mathbb{R}^N. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide non-negativity of the solutions and uniform control of the total mass. The diffusion operators are of type ui↦di(−Δ)suiu_i\mapsto d_i(-\Delta)^s u_i where 0<s<10<s<1. Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type ui↦−diΔuiu_i\mapsto -d_i\Delta u_i. On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case s=1s=1

    Local existence of solutions and comparison principle for initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory

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    We consider an initial value problem for a nonlinear parabolic equation with memory under nonlinear nonlocal boundary condition. In this paper we study classical solutions. We establish the existence of a local maximal solution. It is shown that under some conditions a supersolution is not less than a subsolution. We find conditions for the positiveness of solutions. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem

    A doubly nonlinear fractional diffusive equation

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    This thesis focuses on the `doubly nonlinear fractional diffusive equation', a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional p-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in Euclidean space and composed with a power-like function. We also include a Lipschitz perturbation and a forcing term depending on space and time. We first generalize the nonlinear term u^m, replacing this by a continuous, strictly increasing function. Here we establish well-posedness in L1 in the sense of mild solutions and a comparison principle. For domains with finite measure and with restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong and distributional. We then consider the power-like case where we obtain further regularity properties. In particular, we have an Ll-L∞ regularizing effect for mild solutions (and therefore also for strong solutions), also known as ultracontractivity. We further obtain derivative and energy estimates for this problem. Using these, we extend the previous strong regularity result to obtain strong distributional solutions on general open domains with initial data in L1. Moreover, we prove local and global Hölder continuity results in restricted cases as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation. We finally restrict to the doubly nonlinear fractional diffusive equation without forcing terms, where we investigate self-similarity properties and, in particular, the asymptotic behaviour of solutions for large times. The main result in this case is the existence of Barenblatt solutions. However, in finding these we also prove an Aleksandrov symmetry principle for solutions and estimate solutions by global bounding functions which are integrable in space
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