top of page

Partial Differential Equations

PDEs arising in a majority of real-world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, the so-called nonlinear diffusion equation. In a series of publications a general theory of the nonlinear degenerate and/or singular parabolic equations in general non-smooth domains under the minimal regularity assumptions on the boundary was developed. This development was motivated with numerous applications, including flow in porous media, heat conduction in a plasma, free boundary problems with singularities, etc. Related publications are listed below.

Partial Differential Equations: What We Do

One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a deep connection between this problem and the delicate problem of asymptotics of the corresponding Wiener processes. In the paper

 a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation was proved. In addition, an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation is established. In fact, for the characteristic top boundary point of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovsky test for the local asymptotics of the multi-dimensional Brownian motion trajectories:

 In the case when symmetric rotational boundary surfaces extend to t=-∞, the regularity of the point at ∞ precisely characterize the uniqueness of the bounded solutions. Geometric necessary and sufficient condition for the regularity of ∞ was proved. In the probabilistic context the result coincides with the Kolmogorov-Petrovsky test for the asymptotics of the multi-dimensional Brownian motion trajectories at infinity. Related publications are listed below.

Partial Differential Equations: Welcome
bottom of page