Advection-diffusion equations with random coefficientson evolving hypersurfaces

We present the analysis of advection-diffusion equations with random coefficients on moving hypersurfaces. We define weak and strong material derivative, that take into account also the spacial movement. Then we define the solution space for these kind of equations, which is the Bochner-type space of random functions defined on moving domain. Under suitable regularity assumptions we prove the existence and uniqueness of solutions of the concerned equation, and also we give some regularity results about the solution.

[1]  C. M. Elliott,et al.  Computation of geometric partial differential equations and mean curvature flow , 2005, Acta Numerica.

[2]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Alan Demlow,et al.  Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces , 2009, SIAM J. Numer. Anal..

[5]  Charles M. Elliott,et al.  L2-estimates for the evolving surface finite element method , 2012, Math. Comput..

[6]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[7]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[8]  Omar M. Knio,et al.  Spectral Methods for Uncertainty Quantification , 2010 .

[9]  Charles M. Elliott,et al.  An abstract framework for parabolic PDEs on evolving spaces , 2014, 1403.4500.

[10]  Claude Jeffrey Gittelson,et al.  Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* , 2011, Acta Numerica.

[11]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[12]  Ralf Forster,et al.  On the stochastic Richards equation , 2011 .

[13]  Christoph Schwab,et al.  Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs I: Analytic regularity and gpc-approximation ! , 2010 .

[14]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[15]  Hongkai Zhao,et al.  An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface , 2003, J. Sci. Comput..

[16]  J. Esquena.  Current Opinion in Colloid and Interface Science , 2016 .

[17]  Charles M. Elliott,et al.  Finite element methods for surface PDEs* , 2013, Acta Numerica.

[18]  Stanley Osher,et al.  A Framework for Solving Surface Partial Differential Equations for Computer Graphics Applications , 2000 .

[19]  Martin Berzins,et al.  A computational model for organism growth based on surface mesh generation , 2003 .

[20]  J. Lowengrub,et al.  A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant , 2004 .

[21]  C. J. Gittelson.  STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM , 2010 .

[22]  Stefano Soatto,et al.  Region-Based Segmentation on Evolving Surfaces with Application to 3D Reconstruction of Shape and Piecewise Constant Radiance , 2004, ECCV.

[23]  Giovanni Peccati,et al.  Mean-square continuity on homogeneous spaces of compact groups , 2012 .

[24]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[25]  Robert Scheichl,et al.  Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods , 2013, SIAM J. Numer. Anal..

[26]  Julia Charrier,et al.  Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients , 2012, SIAM J. Numer. Anal..

[27]  Norman R. Morrow,et al.  Recovery of oil by spontaneous imbibition , 2001 .

[28]  Charles M. Elliott,et al.  Finite elements on evolving surfaces , 2007 .

[29]  Catherine E. Powell,et al.  An Introduction to Computational Stochastic PDEs , 2014 .

[30]  Julia Charrier,et al.  Numerical Analysis of the Advection-Diffusion of a Solute in Porous Media with Uncertainty , 2015, SIAM/ASA J. Uncertain. Quantification.

[31]  C. M. Elliott,et al.  Surface Finite Elements for Parabolic Equations , 2007 .

[32]  M. Vierling,et al.  Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control – theory and numerical realization , 2014 .

[33]  Rob P. Stevenson,et al.  Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..

[34]  H. Stone.  A simple derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface , 1990 .

[35]  Stig Larsson,et al.  Quasi-optimality of Petrov-Galerkin discretizations of parabolic problems with random coefficients , 2016, 1604.06611.

[36]  Charles M. Elliott,et al.  On some linear parabolic PDEs on moving hypersurfaces , 2014, 1412.1624.