Partial Differential Equations Seminar

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Past events in this series
16 October 2017
16:00
Abstract

R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes showed in their celebrated paper from 1993 that numerous compensated compactness quantities such as Jacobians of mappings in $W^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong the real-variable Hardy space $\mathcal{H}^1(\mathbb{R}^n)$. They proceeded to ask what is the exact range of these nonlinear quantities and in particular whether the Jacobian operator $J$ maps $W^{1,2}(\mathbb{R}^2,\mathbb{R}^2)$ onto $\mathcal{H}^1(\mathbb{R}^2)$.

I present the proof of my recent result that $J \colon W^{1,n}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^1(\mathbb{R}^n)$ is non-surjective for every $n \ge 2$. The surjectivity question is still open when the domain of definition of $J$ is the inhomogeneous Sobolev space $\dot{W}^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$. I also shortly discuss my work on T. Iwaniec's conjecture from 1997 which states that for every $n \ge 2$ and $p \in [1,\infty[$ the operator $J \colon W^{1,np}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^p(\mathbb{R}^n)$ has a continuous right inverse.

  • Partial Differential Equations Seminar
30 October 2017
16:00
Igor Pazanin
Abstract

It is well-known that only a limited number of the fluid flow problems can be solved (or approximated) by the solutions in the explicit form. To derive such solutions, we usually need to start with (over)simplified mathematical models and consider ideal geometries on the flow domains with no distortions introduced. However, in practice, the boundary of the fluid domain can contain various small irregularities (rugosities, dents, etc.) being far from the ideal one. Such problems are challenging from the mathematical point of view and, in most cases, can be treated only numerically. The analytical treatments are rare because introducing the small parameter as the perturbation quantity in the domain boundary forces us to perform tedious change of variables. Having this in mind, our goal is to present recent analytical results on the effects of a slightly perturbed boundary on the fluid flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter $\varepsilon$ and arbitrary smooth function. The porous medium flow is described by the Darcy-Brinkman model which can handle the presence of a boundary on which the no-slip condition for the velocity is imposed. Using asymptotic analysis with respect to $\varepsilon$, we formally derive the effective model in the form of the explicit formulae for the velocity and pressure. The obtained asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution. We will also address the problem of the solute transport through a semi-in finite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel's walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number-regime is derived using asymptotic analysis with respect to $\varepsilon$. The obtained mathematical model clearly indicates the influence of the porous medium, chemical reaction and boundary distortion on the effective flow.

This is a joint work with Eduard Marušić-Paloka (University of Zagreb).

  • Partial Differential Equations Seminar
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