Past PDE CDT Lunchtime Seminar

19 November 2020
12:00
Ph.D. Gianmarco Sperone
Abstract

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.


The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).

 

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  • PDE CDT Lunchtime Seminar
5 November 2020
12:00
Abstract

Solving kinetic or related models with high-dimensional random parameters has been a challenging problem. In this talk, we will discuss how to employ the bi-fidelity stochastic collocation and choose efficient low-fidelity models in order to solve a class of multi-scale kinetic equations with uncertainties, including the Boltzmann equation, linear transport and the Vlasov-Poisson equation. In addition, some error analysis for the bi-fidelity method based on these PDEs will be presented. Finally, several numerical examples are shown to validate the efficiency and accuracy of the proposed method.

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  • PDE CDT Lunchtime Seminar
22 October 2020
12:00
André Guerra / Lukas Koch
Abstract

I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous and weak* stable. An example of such a PDE is the Jacobian equation. I will discuss the consequences of such a result for the Jacobian and its relevance towards an answer to a long-standing problem due to Coifman, Lions, Meyer and Semmes. This is based on joint work with Lukas Koch and Sauli Lindberg.

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I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

 

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  • PDE CDT Lunchtime Seminar
15 October 2020
12:00
James Kohout / Jeremy Wu
Abstract

In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk we will introduce this problem of uniqueness of asymptotic limits and discuss joint work with Melanie Rupflin and Peter M. Topping in which we examined the situation for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface.

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The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with Jose Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

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  • PDE CDT Lunchtime Seminar
18 June 2020
12:00
Sebastian Schwarzacher
Abstract

I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.

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  • PDE CDT Lunchtime Seminar
11 June 2020
12:00
Abstract

In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

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  • PDE CDT Lunchtime Seminar
28 May 2020
15:00
Abstract

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

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  • PDE CDT Lunchtime Seminar
7 May 2020
12:00
Cristiana De FIlippis
Abstract

Non-uniformly elliptic functionals are variational integrals like
\[
(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,
\]
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$
\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\  \partial_{z}^{2} F(x,z)} $$
may blow up as $|z|\to \infty$. 
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$
w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x
$$
or
$$
w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.
$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).

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  • PDE CDT Lunchtime Seminar
5 March 2020
12:00
Lenka Slavíková
Abstract

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.

  • PDE CDT Lunchtime Seminar

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