The Parabolic Case: Two Coinciding Characteristics

2020 ◽  
pp. 160-178
Author(s):  
Otto D. L. Strack
Keyword(s):  
Parabolic Case

Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random source

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roberto Díaz-Adame ◽  
Silvia Jerez
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Splitting Method ◽  
Diffusion Equations ◽  
Hyperbolic Conservation ◽  
Convection Diffusion ◽  
Parabolic Case ◽  
Weak Entropy Solution ◽  
Degenerate Parabolic ◽  
Time Splitting ◽  
Operator Scheme

AbstractIn this paper we propose a time-splitting method for degenerate convection-diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in $\begin{array}{} \displaystyle L^p_{loc} \end{array}$ of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a fluid flow application in porous media.

External optimal control of fractional parabolic PDEs

2020 ◽  
Vol 26 ◽  
pp. 20 ◽  
Author(s):  
Harbir Antil ◽  
Deepanshu Verma ◽  
Mahamadi Warma
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Convergence Rates ◽  
Complete Analysis ◽  
Very Weak Solutions ◽  
Parabolic Pdes ◽  
Nonlocal Models ◽  
Parabolic Case ◽  
Potential Benefits ◽  
Control Source

In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

scholarly journals SOME LINEAR SPDEs DRIVEN BY A FRACTIONAL NOISE WITH HURST INDEX GREATER THAN 1/2

2012 ◽  
Vol 15 (04) ◽  
pp. 1250023 ◽  
Author(s):  
RALUCA M. BALAN
Keyword(s):  
Stable Process ◽  
Sufficient Conditions ◽  
Hyperbolic Type ◽  
Hurst Index ◽  
Intersection Local Time ◽  
Fractional Noise ◽  
Spatial Operator ◽  
Parabolic Case ◽  
Spatially Homogeneous ◽  
Necessary And Sufficient

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear stochastic partial differential equations (spde's) of parabolic and hyperbolic type. These equations rely on a spatial operator [Formula: see text] given by the L2-generator of a d-dimensional Lévy process X = (Xt)t≥0, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index H > 1/2. As an application, we consider the case when X is a β-stable process, with β ∈ (0, 2]. In the parabolic case, we develop a connection with the potential theory of the Markov process [Formula: see text] (defined as the symmetrization of X), and we show that the existence of the solution is related to the existence of a "weighted" intersection local time of two independent copies of [Formula: see text].

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

2005 ◽  
Vol 176 (3) ◽  
pp. 415-453 ◽  
Author(s):  
Fuensanta Andreu ◽  
Vicent Caselles ◽  
José M. Mazón
Keyword(s):  
Quasilinear Equation ◽  
Parabolic Case ◽  
Strongly Degenerate

scholarly journals Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 1-26 ◽  
Author(s):  
Raffaele Folino
Keyword(s):  
Initial Velocity ◽  
Numerical Experiments ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Slow Motion ◽  
Cahn Equation ◽  
Parabolic Case ◽  
Appl Math ◽  
One Space Dimension

The aim of this paper is to prove that, for specific initial data [Formula: see text] and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval [Formula: see text] shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990) 983–997], if [Formula: see text] is the diffusion coefficient, we show that in a time scale of order [Formula: see text] nothing happens and the solution maintains the same number of transitions of its initial datum [Formula: see text]. The novelty consists mainly in the role of the initial velocity [Formula: see text], which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.

scholarly journals Regularity for the fully nonlinear parabolic thin obstacle problem

2021 ◽  
pp. 2150011
Author(s):  
Georgiana Chatzigeorgiou
Keyword(s):  
Parabolic Equation ◽  
Viscosity Solution ◽  
Nonlinear Parabolic Equation ◽  
Obstacle Problem ◽  
Elliptic Case ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Parabolic Case ◽  
Thin Obstacle

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

Homogenization of Dirichlet parabolic systems with variable monotone operators in general perforated domains

2003 ◽  
Vol 133 (6) ◽  
pp. 1231-1248 ◽  
Author(s):  
Carmen Calvo-Jurado ◽  
Juan Casado-Díaz
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Dirichlet Boundary ◽  
Monotone Operators ◽  
Parabolic Systems ◽  
Limit Problem ◽  
Dirichlet Boundary Conditions ◽  
Perforated Domains ◽  
Parabolic Case

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

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