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Parabolic equations with rough data

Herbert Koch, Tobias Lamm (2015)

Mathematica Bohemica

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local...

Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

Vagif S. Guliyev, Mehriban N. Omarova (2016)

Open Mathematics

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space [...] W˙2,1p,φ(Q,ω) W ˙ 2 , 1 p , ϕ Q , ω .

Parabolic potentials and wavelet transforms with the generalized translation

Ilham A. Aliev, Boris Rubin (2001)

Studia Mathematica

Parabolic wavelet transforms associated with the singular heat operators - Δ γ + ∂ / ∂ t and I - Δ γ + ∂ / ∂ t , where Δ γ = ∑ k = 1 n ∂ ² / ∂ x ² k + ( 2 γ / x ₙ ) ∂ / ∂ x ₙ , are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderón reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations

Yan Xie, Jie Cheng, Benzhuo Lu, Linbo Zhang (2013)

Molecular Based Mathematical Biology

rithms for solving the 3D electro-diffusion equations such as the Poisson-Nernst-Planck equations and the size-modified Poisson-Nernst-Planck equations in simulations of biomolecular systems in ionic liquid. A set of transformation methods based on the generalized Slotboom variables is used to solve the coupled equations. Calculations of the diffusion-reaction rate coefficients, electrostatic potential and ion concentrations for various systems verify the method’s validity and stability. The iterations...

Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation

Minh-Binh Tran (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with...

Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE

E. Grenier, V. Louvet, P. Vigneaux (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases...

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