A Boundary Estimate for Quasi-Linear Diffusion Equations

Ugo Gianazza

Institution:

Università degli Studi di Pavia (Italy)

Time:

Friday, April 2, 2021 - 3:00pm to 4:00pm

Location:

Zoom

Let $E$ be an open set in $\mathbb{R}^N$, and for $T>0$ let $E_T$ denote the cylindrical domain $E\times[0,T]$. We consider quasi-linear, parabolic partial differential equations of the form $$u_t-\operatorname{div}\textbf{A}(x,t,u, Du) = 0\quad \text{ weakly in }\> E_T,$$ where the function $\textbf{A}(x,t,u,\xi)\colon E_T\times\mathbb{R}^{N+1}\to\mathbb{R}^N$ is assumed to be measurable with respect to $(x, t) \in E_T$ for all $(u,\xi)\in\mathbb{R}\times\mathbb{R}^N$, and continuous with respect to $(u,\xi)$ for a.e.~$(x,t)\in E_T$. Moreover, we assume the structure conditions $$\begin{cases} \textbf{A}(x,t,u,\xi)\cdot \xi\geq C_0|\xi|^p,&\\ \textbf{A}(x,t,u,\xi)|\leq C_1|\xi|^{p-1},& \end{cases}$$ for a.e. $(x,t)\in E_T$, $\forall\,u\in\mathbb{R},\,\forall\xi\in\mathbb{R}^N$, where $C_0$ and $C_1$ are given positive constants, and we take $p>1$. We consider a boundary datum $g$ with $$\begin{cases} g\in L^p\big(0,T;W^{1,p}( E)\big),&\\ g \text{ continuous on}\ \overline{E}_T\ \text{with modulus of continuity }\ \omega_g(\cdot),& \end{cases}$$ and we are interested in the boundary behavior of solutions to the Cauchy-Dirichlet problem $$\begin{cases} u_t-\operatorname{div}\textbf{A}(x,t,u, Du) = 0&\text{ weakly in }\> E_T,\\ u(\cdot,t)\Big|_{\partial E}=g(\cdot,t)&\text{ a.e. }\ t\in(0,T],\\ u(\cdot,0)=g(x,0),& \end{cases}$$ with $g$ as above. We do not impose any {\em a priori} requirements on the boundary of the domain $E\subset\mathbb{R}^N$, and we provide an estimate on the modulus of continuity at a boundary point in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity. The results depend on the value of $p$, namely whether $1<p\le\frac{2N}{N+1}$, $\frac{2N}{N+1}<p<2$, $p\ge2$.

This is a joint work with Naian Liao (Salzburg University, Austria) and Teemu Lukkari (Aalto University, Finland).

The Theorem of Characterization of the Strong Maximum Principle

Speaker:

Julián Lopez-Gomez

Institution:

Universidad Complutense de Madrid (Spain)

Time:

Friday, May 7, 2021 - 3:00pm to 4:00pm

Location:

Zoom

This talk begins with a discusion of the classical minimum principle of Hopf and the boundary lemma of Hopf—Oleinik to infer from them the generalized minimum principle of Protter—Weinberger. Then, a technical device of Protter and Weinberger is polished and sharpened to get a fundamental theorem on classification of supersolutions which provides with the theorem of characterization of the strong maximum principle of Amann and Molina-Meyer together with the speaker. Finally, some important applications of this theorem are discussed. The talk adopts the general patterns of Chapters 1, 2, 6 and 7 of the book on Elliptic Operators of the speaker.

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Stability of Self-Similarity Solutions of Surface Diffusion

Aaron Yip

Institution:

Purdue University

Time:

Friday, May 14, 2021 - 3:00pm to 5:00pm

Location:

Zoom

Surface diffusion (SD) is a curvature driven flow where a (hyper-)surface evolves by the surface Laplacian of its mean curvature. It is a fourth order parabolic equation. Compared with its second order counterpart, motion by mean curvature (MMC), for which maximum principle is available, much less is known for SD. I will present a stability result for self-similarity solutions. Though the approach is based on linearization, and it only works for the evolution of graphs, it is quite robust and works for both MMC and SD. I will also present an attempt to analyze the pinch-off phenomena for axisymmetric surfaces. The former is based on a joint work with Hengrong Du and the later with Gavin Glenn.

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Fluid Flow in General Domains

Jürgen Saal

Institution:

Heinrich-Heine-Universität Düsseldorf (Germany)

Time:

Friday, March 5, 2021 - 3:00pm to 4:00pm

Location:

Zoom
(Incompressible) Fluid flow in a domain is described by the fundamental Stokes (linear) and Navier-Stokes (nonlinear) equations. The Helmholtz decomposition into solenoidal and gradient fields serves as a helpful tool to analyze these systems. It has been an open question for some decades, whether the existence of the Helmholtz decomposition (which is equivalent to weak well-posedness of the Neumann problem) is necessary for well-posedness of Stokes and Navier-Stokes equations in the $L^q$-setting for $q\in(1,\infty)$. Note that by a classical result of Bogovski\u{i} and Maslennikova there are uniformly smooth domains, so-called non-Helmholtz domains, such that the Helmholtz decomposition does not exist. In my talk, I intent to present positive and negative results on well-posedness of the Stokes and Navier-Stokes equations in $L^q$ for a large class of uniform $C^{2,1}$-domains. In particular, classes of non-Helmholtz domains are addressed. This will include a comprehensive answer to the open question for the case of partial slip type boundary conditions.
The project is a joint work with Pascal Hobus.
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Long-time dynamics of some nonlocal diffusion models with free boundary

Yihong Du

Institution:

University of New England (Australia)

Time:

Friday, January 15, 2021 - 3:00pm to 4:00pm

Propagation has been modelled by reaction-diffusion equations since the pioneering works of Fisher and Kolmogorov-Peterovski-Piskunov (KPP). Much new developments have been achieved in the past a few decades on the modelling of propagation, with traveling wave and related solutions playing a central role. In this talk, I will report some recent results obtained with several collaborators on some reaction-diffusion models with free boundary and "nonlocal diffusion", which include the Fisher-KPP equation (with free boundary) and two epidemic models.  A key feature of these problems is that the propagation may or may not be determined by traveling wave solutions.  There is a threshold condition on the kernel functions which determines whether the propagation has a finite speed or infinite speed (known as accelerated spreading). For some typical kernel functions, we obtain sharp estimates of the spreading speed (whether finite or infinite).

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The total variation flow in metric random walk spaces

José Mazón

Institution:

Universitat de Valencia (Spain)

Time:

Friday, April 9, 2021 - 3:00pm to 4:00pm

Location:

Zoom

Our aim is to study the Total Variation Flow (TVF) in metric random walk spaces (MRWS) which include as particular cases: the TVF on locally finite weighted connected graphs. We introduce the concepts of perimeter and mean curvature for subsets of a MRWS. After proving the existence and uniqueness of solutions of the TVF, we study the asymptotic behaviour of those solutions, and for such aim we establish some inequalities of Poincar ́e type. Furthermore, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the 1-Laplacian operator. In connection with the Cheeger cut problem we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.

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A new variational principle with applications in partial differential equations and Analysis

Speaker:

Abbas Momeni (Moameni)

Institution:

Carleton University

Time:

Friday, December 4, 2020 - 3:00pm to 4:00pm

Location:

Zoom

The purpose of this presentation is to introduce a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.

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Recent Advances concerning the Navier-Stokes and Euler Equations

Edriss Titi

Institution:

University of Cambridge - Texas A&M University - Weizmann Institute of Science

Time:

Friday, October 30, 2020 - 3:00pm to 4:00pm

Location:

Zoom

In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness and the effect of physical boundaries on the potential formation of singularity. In addition, I will present a blow-up criterion based on a class of inviscid regularization for these equations.

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Aggregation with intrinsic interactions on Riemannian manifolds

Razvan Fetecau

Institution:

Simon Fraser University

Time:

Friday, October 23, 2020 - 3:00pm to 4:00pm

Location:

Zoom

We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group SO(3)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

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Moving Boundary Problems and Medicine

Sunčica Čanić

UC Berkeley

Time:

Friday, December 11, 2020 - 3:00pm to 4:00pm

Location:

Zoom

With the recent developments of new technologies in biomedical engineering and medicine, the need for new mathematical and numerical methdologies to aid these developments has never been greater. In particular, the design of next generation drug-eluting stents for the treatment of coronary artery disease, and the design of an implantable bioartificial pancreas for the treatment of Type 1 diabetes, hinge on the development of mathematical and computational techniques for solving nonlinear moving boundary problems. In this talk we present recent developments and open problems in the mathematical theory of nonlinear moving boundary problems describing the interaction between viscous, incompressible fluids, and elastic and poroelastic structures. Applications of the mathematical results to coronary angioplasty with stenting, and to bioartificial pancreas design will be shown.

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