Department of Mathematics, Purdue University
PDE Seminar

FALL 2023

Talks in FALL 2023 will take place at 2:00-3:00pm, Thursday, MATH 731. It is a hybrid format with majority of talks being given in persons, along with some virtual talks as well. If you have any questions about a particular seminar, please contact the organizers at: or

September 7 Luigi De Rosa, University of Basel, Switzerland (Virtual)
TITLE: Intermittency and lower dimensional dissipation in fluid dynamics
ABSTRACT: In the context of incompressible fluids, there is a quite strong empirical evidence of a lower dimensional accumulation of energy dissipation in the infinite Reynolds number limit. This phenomenon is nowadays known as "intermittency" and it is intrinsically linked to non-homogeneous regularity exponents observed in the velocity field. In this talk I will describe a couple of possible ways to put the study of intermittent flows in a rigorous mathematical setting. This builds on various refinements of the theory of divergence measure fields, together with different notions of dimensions. The main difference in the choice of the notion of dimension is the possibility to study dissipations which happen to concentrate on dense sets.
September 14 Thomas Schmidt, University of Hamburg (Virtual)
TITLE: Perimeter functionals with measure data
ABSTRACT : The talk is concerned with perimeter functionals $\mathscr{P}_{\mu_+,\mu_-}$ given by \[ \mathscr{P}_{\mu_+,\mu_-}[A]:=\mathrm{P}(A)+\mu_+(A^1)-\mu_-(A^+) \] on measurable sets $A\subset{\mathbb R}^n$ of finite volume $|A|<\infty$ and finite perimeter $\mathrm{P}(A)<\infty$, where the fixed non-negative Radon measures $\mu_\pm$ on $\mathds{R}^n$ are (necessarily) evaluated on a measure-theoretic interior $A^1$ and a measure-theoretic closure $A^+$ of $A$. It will be shown that various existence results for minimizers of the functional $\mathscr{P}_{\mu_+,\mu_-}$ can be obtained under a new and optimal condition on $\mu_\pm$, called the small-volume isoperimetric condition. Moreover, it will be illustrated at hand of exemplary configurations that this condition admits a wide class of $(n{-}1)$-dimensional measures and that it is decisive for lower semicontinuity with cancellation-compensation effects between the perimeter and measure terms. The long-term goal of these considerations is to extend the variational approach to prescribed-mean-curvature boundaries $\partial A$ in the spirit of Caccioppoli, De Giorgi, Miranda, Massari from $\mathrm{L}^1$ mean curvature to mean curvature given by a possibly lower-dimensional signed measure.
September 21 Jay Bang, University of Texas at San Antonio
TITLE: Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its Applications
ABSTRACT : Solutions with scaling-invariant bounds such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. I proved with collaborators that any steady solution satisfying $|u(x)|\leq C/|x|$ for any constant $C$ in $\mathbb{R}^n\setminus \{0\}$ with $n\geq 4$, must be zero. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers. The proof is pretty elementary and short. We do not assume any smallness or self-similarity. By using a blow-up method, this theorem also help to remove a class of singularities of solutions and give the optimal asymptotic behaviors of solutions at infinity in the exterior domains. This is a joint work with Changfeng Gui, Hao Liu, Yun Wang, and Chunjing Xie.
September 28 Theodore Drivas, Stony Brook University (Virtual)
TITLE: The Feynman-Lagerstrom criterion for boundary layers
ABSTRACT: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations.
October 5 Yannick Sire, Johns Hopkins University
TITLE: Harmonic maps between singular spaces
ABSTRACT : After reviewing briefly the classical theory of harmonic maps between smooth manifolds, I will describe some recent results related to harmonic maps with free boundary, emphasizing on two different approaches based on recent developments by Da Lio and Riviere. This latter approach allows in particular to give another formulation which is well-suited for such maps between singular spaces. After the works of Gromov, Korevaar and Schoen, harmonic maps between singular spaces have been instrumental to investigate super-rigidity in geometry. I will report on recent results where we introduce a new energy between singular spaces and prove a version of Takahashi’s theorem (related to minimal immersions by eigenfunctions) on RCD spaces
October 12 (NO SEMINAR)
October 16, 1:30-2:30pm (SPECIAL TIME) Stefania Patrizi, University of Texas at Austin
TITLE: Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model
ABSTRACT : We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations, a popular model describing the evolution of the density of parallel straight dislocation lines. One of the main difficulties is that we allow dislocations to have opposite orientation and so we have to deal with collisions of them. This is a joint work with Tharathep Sangsawang.
October 26 Zhiyuan Geng, Purdue University (Virtual)
TITLE: On the triple junction problem without symmetry hypotheses
ABSTRACT: For the scalar two-phase (elliptic) Allen-Cahn equation, there is a rich literature dedicated to a celebrated conjecture of De-Giorgi, which establishes a connection between the diffuse interface and minimal surfaces. However, for the case of three or more equally preferred phases, a vector order parameter is required. In this scenario, the structure of the diffuse interface is analogous to a singular minimal cone. These solutions are studied mostly under an assumption of equivariance of the potential and the solution. In this talk, I will present the joint work with Nicholas Alikakos on the Allen-Cahn equation with a potential vanishing at exactly three points (energy wells). Our work involves constructing a locally minimizing entire solution that possesses a triple junction structure. We estimate the location and size of the diffuse interface by utilizing an energy upper/lower bound. Additionally, we investigate the asymptotic behavior at infinity by examining the blow-down limit. We do not impose any symmetry hypothesis on the solution or potential.
November 2 Jincheng Yang, University of Chicago
TITLE: Higher regularity and trace estimates for Navier-Stokes equation
ABSTRACT : We derive several nonlinear a priori trace estimates for the 3D incompressible Navier-Stokes equation. They recover and extend the current picture of higher derivative estimates in the mixed norm. The main ingredient is the blow-up method and a novel averaging operator, which could apply to PDEs with scaling invariance and quantitative one-scale ε-regularity.
November 9 Andre Schlichting, University of Munster, Germany (Virtual)
TITLE: Mean-field PDEs on graphs and their continuum limit
ABSTRACT : This talk reviews some recent results on nonlocal PDEs describing the evolution of a density on graph structures. These equations can arise from mean-field interacting jump dynamics, but also from applications in the data science field, or they can also be obtained by a numerical discretization of a continuum problem. We also show how those equations are linked to their continuous counterpart in suitable local limits. Joint works with Antonio Esposito, Georg Heinze, Anastasiia Hraivoronska and Oliver Tse.
November 16 Matt Novack, Purdue University
TITLE: Weak kinetic shock solutions to the Landau equation
ABSTRACT : Compressible fluids are known to form shock waves, which can be represented by discontinuous solutions of the compressible Euler equations. However, physical shocks are actually continuous and in certain regimes can be represented by a smooth shock profile. In this talk, I will discuss a construction of weak shock profiles which solve the kinetic Landau equation. This is based on joint work with Dallas Albritton (Wisconsin) and Jacob Bedrossian (UCLA).
November 23 No Seminar (Thanksgiving Holiday)

November 30 Andrea Pinamonti, Universita di Trento, Italy (Virtual)
TITLE: Variational solutions to the $\infty-$Poisson equation with respect to Hörmander vector fields
ABSTRACT : In this talk we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carath\'eodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi \cite{BDM} and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation. The talk is based on a joint paper with L. Capogna, G. Giovannardi and S. Verzellesi.