# An adaptive multiresolution method for parabolic PDEs.

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We present a new adaptive numerical scheme for solving parabolic PDEs in

cartesian geometry. Applying a finite volume discretization with explicit

time integration, both of second order, we employ a fully adaptive

multiresolution scheme to represent the solution on locally refined nested

grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption

strategy to advance the grid in time and to follow the time evolution of

the solution directly explaoits the multiresolution representation.

Applying this new method to several test probelms in one, two and three

space dimensions, like convection-diffucion, viscous Burgers and

reaction-diffusion equations, we show its second order accuracy and

demonstrate its computational efficiency.

This work is joint work with Olivier Roussel.

# Dynamics of a Nonlinear Convection-diffusion Equation in Multindimensional Bounded Domains. (After A. Hill and E. Suli). Lecture 1.

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# Dynamics of a Nonlinear Convection-diffusion Equation in Multindimensional Bounded Domains. (After A. Hill and E. Suli). Lecture 2.

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# Dynamics of a Nonlinear Convection-diffusion Equation in Multindimensional Bounded Domains. (After A. Hill and E. Suli). Lecture 2.

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# On the Regularity Conditions for the Navier-Stokes and the Related Equations

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In this talk I present my recent results on the regularity conditions for a solution to the 3D Navier-Stokes equations with powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. For the proof of the we exploit geometric properties of the vortex stretching term as well as the estimate using the Triebel-Lizorkin type of norms.