PDE Real Analysis Seminar

Seminar information archive ~01/15Next seminarFuture seminars 01/16~

Date, time & place Tuesday 10:30 - 11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Seminar information archive


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Alex Mahalov (Arizona State University)
Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)
[ Abstract ]
We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Felix Schulze (University College London)
Optimal isoperimetric inequalities for surfaces in any codimension
in Cartan-Hadamard manifolds (English)
[ Abstract ]
Let $(M^n,g)$ be simply connected, complete, with non-positive sectional
curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area
minimising 3-current such that $\partial S = \Sigma$. We use a weak mean
curvature flow, obtained via elliptic regularisation, starting from
$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric
inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the
optimal estimate in case the sectional curvatures of $M$ are bounded from
above by $\kappa < 0$ and characterise the case of equality. The proof
follows from an almost monotonicity of a suitable isoperimetric
difference along the approximating flows in one dimension higher.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Kaj Nyström (Uppsala University)
Boundary value problems for parabolic equations with measurable coefficients (English)
[ Abstract ]
In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Hoài-Minh Nguyên (École Polytechnique Fédérale de Lausanne)
Some perspectives on negative index materials (English)
[ Abstract ]
Negative index materials (NIMs) are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1964. The existence of NIMs was confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications. NIMs have attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties. Mathematically, the study of NIMs faces two difficulties. First, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Second, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this talk I will discuss various mathematics techniques used to understand various applications of NIMs such as cloaking and superlensing and to develop new designs for them.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Tsuyoshi Yoneda (University of Tokyo)


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Nader Masmoudi (Courant Institute, NYU)
On the stability of the 3D Couette Flow (English)
[ Abstract ]
We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yannick Sire (Johns Hopkins University)
De Giorgi conjecture and minimal surfaces for integro-differential operators (English)
[ Abstract ]
I will review the classical De Giorgi conjecture and its link with minimal surfaces. Then I will move on recent results for flatness of level sets of solutions of semi linear equations involving anomalous diffusion. First I will deal with the fractional laplacian; second with quite general integral operators in 2 dimensions.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Nam Quang Le (Indiana University)
Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)
[ Abstract ]
The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.


10:30-11:30   Room #268 (Graduate School of Math. Sci. Bldg.)
Nguyen Cong Phuc (Louisiana State University)
The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)
[ Abstract ]
In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.
This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.
[ Reference URL ]


10:20-11:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Piotr Rybka (University of Warsaw)
Special cases of the planar least gradient problem (English)
[ Abstract ]
We study the least gradient problem in two special cases:
(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or
(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.


14:20-15:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Amru Hussein (TU Darmstadt)
Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)
[ Abstract ]
Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.


12:10-12:50   Room #056 (Graduate School of Math. Sci. Bldg.)
Elio Espejo (National University of Colombia)
The role of convection in some Keller-Segel models (English)
[ Abstract ]
An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.


11:20-12:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Monika Muszkieta (Wroclaw University of Science and Technology)
The total variation flow in $H^{−s}$ (English)
[ Abstract ]
In the talk, we consider the total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.
This is a join work with Y. Giga.


15:00-16:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Elijah Liflyand (Bar-Ilan University, Israel)
Fourier transform versus Hilbert transform (English)
[ Abstract ]
We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.
1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let $\{a_k\},$ $k=0,1,2...,$ be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function $f:\mathbb T=[-\pi,\pi)\to \mathbb C,$ that is $\sum |a_k|<\infty.$ Under which conditions on $\{a_k\}$ the re-expansion of $f(t)$ ($f(t)-f(0)$, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?
We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.
2. The following result is due to Hardy and Littlewood: If a (periodic) function $f$ and its conjugate $\widetilde f$ are both of bounded variation, their Fourier series converge absolutely.
We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.
3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.
There are multidimensional generalizations of these results.
[ Reference URL ]


16:00-17:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Boris Khesin (University of Toronto)
Fluids, vortex membranes, and skew-mean-curvature flows (English)
[ Abstract ]
We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Salomé Oudet (University of Tokyo)
Hamilton-Jacobi equations for optimal control on 2-dimensional junction (English)
[ Abstract ]
We are interested in infinite horizon optimal control problems on 2-dimensional junctions (namely a union of half-planes sharing a common straight line) where different dynamics and different running costs are allowed in each half-plane. As for more classical optimal control problems, ones wishes to determine the Hamilton-Jacobi equation which characterizes the value function. However, the geometric singularities of the 2-dimensional junction and discontinuities of data do not allow us to apply the classical results of the theory of the viscosity solutions.
We will explain how to skirt these difficulties using arguments coming both from the viscosity theory and from optimal control theory. By this way we prove that the expected equation to characterize the value function is well posed. In particular we prove a comparison principle for this equation.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Hao Wu (Fudan University)
Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flows
[ Abstract ]
In this talk, the general Ericksen-Leslie (E-L) system modelling the incompressible nematic liquid crystal flow will be discussed.
We shall prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients.
In particular, we shall discuss the connection between Parodi's relation and stability of the E-L system.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Tuomo Kuusi (Aalto University)
Nonlocal self-improving properties (English)
[ Abstract ]
The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $W^{1,2}$-Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This is a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Lin Wang (Tsinghua University)
Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)
[ Abstract ]
By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Sumio Yamada (Gakushuin University)
Convex bodies and geometry of some associated Minkowski functionals (日本語)
[ Abstract ]
In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Italo Capuzzo Dolcetta (Università degli Studi di Roma "La Sapienza")
Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)
[ Abstract ]
In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.
We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.
The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.
We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Wojciech Zajączkowski (Institute of Mathematics Polish Academy of Sciences)
Global regular solutions to the Navier-Stokes equations which remain close to the two-dimensional solutions (English)
[ Abstract ]
We consider the motion of the Navier-Stokes equations in a cylinder with the Navier-boundary conditions. First we prove global existence of regular two-dimensional solutions non-decaying in time. Next we show stability of these solutions. In this way we have existence of global regular solutions which remain close to the two-dimensional solutions. We prove the results for nonvanishing external force in time.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Elio Eduardo Espejo (National University of Colombia / Osaka University)
Global existence and asymptotic behavior for some Keller-Segel systems coupled with Navier-Stokes equations (英語)
[ Abstract ]
There are plenty of examples in nature, where cells move in response to some chemical signal in the environment. Biologists call this phenomenon chemotaxis. In my talk I will approach the problem of describing mathematically the phenomenon of chemotaxis when it happens surrounded by a fluid. This is a new research topic bringing the attention of many scientists because it has given rise to many interesting questions having relevance in both biology and mathematics. In particular, I will present some new mathematical models arising from my current research that have given rise to Keller-Segel type systems coupled with Navier-Stokes systems. I will present some results of global existence and asymptotic behavior. Finally I will discuss some open problems.


10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Tsubasa Itoh (Tokyo Institute of Technology)
Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)
[ Abstract ]
In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.
It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.
(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)


16:30-17:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Kentarou Yoshii (Faculty of Science Division I, Tokyo University of Science)
On the abstract evolution equations of hyperbolic type (JAPANESE)
[ Abstract ]
This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

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