Title of Talk：On the Helmholtz decomposition of BMO spaces of vector fields
Speaker：Prof.Yoshikazu Giga (University of Tokyo)
Talk Time：October 11st, 2022, 16:00—17:00 pm (Beijing time) 17:00—18:00 pm (Japan time)
Abstract: The Helmholtz decomposition of vector fields is a fundamental tool for analysis of vector fields especially to analyze the Navier-Stokes equations in a domain. It gives a unique decomposition of a (tangential) vector field defined in a domain of an Euclidean space (or aRiemannian manifold) into a sum of a gradient field and a solenoidal field with supplemental condition like a boundary condition. It is well-known that such decomposition gives an orthogonal decomposition of the space of L^2 vector fields in an arbitrary domain and known as the Weyl decomposition. It is also well-studied that in various domains including the half space, smooth bounded and exterior domain, it gives a topological direct sum decomposition of the space of L^p vector fields for 1 < p < ∞. The extension to the case p=∞ (or p=1) is impossible because otherwise it would imply the boundedness of the Riesz type operator in L^∞ (or L^1) which is absurd.
In this talk, we extend the Helmholtz decomposition in a space of vector fields with bounded mean oscillations (BMO) when the domain of the vector field is a smooth bounded domain in an Euclidean space. There are several possible definitions of a BMO space of vector fields. However, to have a topological direct sum decomposition, it turns out that components of normal and tangential to the boundary should be handled separately.
This decomposition problem is equivalent to solving the Poisson equation with the divergence of the original vector field v as a data with the Neumann data with the normal trace of v. The desired gradient field is the gradient of the solution of this Poisson equation. To solve this problem, we construct a kind of volume potential so that the problem is reduced to the Neumann problem for the Laplace equation. Unfortunately, taking the usual Newton potential causes a problem to estimate the necessary norm so we construct another volume potential based on normal coordinate. We need a trace theorem to control L^∞ norm of the normal trace. This is of independent interest. Finally, we solve the Neumann problem with L^∞ data in a necessary space. The Helmholtz decomposition for BMO vector fields is previously known only in the whole Euclidean space or the half space so this seems to be the first result for a domain with a curved boundary. This is a joint work with my student Z.Gu (University of Tokyo).
Zoom Information
https://zoom.us/j/97445030487?pwd=VEhYQVVMRUlyNjBmTXU1R0crczMydz09
Conference number：974 4503 0487
Passwords：742506
Title of Talk：Motion by crystalline-like mean curvature
Speaker：Prof.Yoshikazu Giga (University of Tokyo)
Talk Time：October 18th, 2022, 16:00—17:00 pm (Beijing time) 17:00—18:00 pm (Japan time)
Abstract:In materials science, it is popular to describe the motion of phase boundaries by anisotropic curvature flows. A crystalline mean curvature flow is a typical example. It is a kind of a gradient flow whose energy functional is singular so the meaning of a solution itself is not clear. A prototype is the total variation flow and its speed of a part of slope zero is determined by some nonlocal quantity.
For a planar motion, a level-set approach for a crystalline curvature flow was established two decades ago. The situation is easier since the expected speed of a flat part called a facet is constant depending on its length. However, for a surface motion, the speed of a facet may not be constant and facet-splitting or facet-bending may occur. If the speed is constant, the facet is often called calibrable and its value must be an anisotropic Cheeger ratio.
In this talk, we survey recent progress of a level-set approach to crystalline mean curvature flows including spatially inhomogeneous driving forces. A key point is how to establish the notion of a solution based on crystalline curvature based on the theory of viscosity solutions. One should be careful to give a “correct” notion so that the unique solution is given by the limit of solutions of approximate problems, especially when there is an inhomogeneous driving force. This talk is based on my recent joint work with N. Požár (Kanazawa University).
Zoom Information:
https://zoom.us/j/94891144904?pwd=WEFFUVFabWxzSzlrODJoYnRWR1A1dz09
Conference number：948 9114 4904
Passwords：778927
Talk Title：The fourth-order total variation flows in R^n
Speaker：Prof.Yoshikazu Giga (University of Tokyo)
Talk Time：October 25th, 2022, 16:00—17:00 pm (Beijing time) 17:00—18:00 pm (Japan time)
Abstract: The total variation flow is by now very popular for image denoising and also in materials science. When we model relaxation of crystal surfaces below the roughening temperature, the fourth-order total variation flow is often used. However, compared with the second-order problem, its analytic property has not been well studied because the definition of the solution itself is not trivial.
In this talk, we define rigorously its solution in the whole Euclidean space R^n. It turns out that if n is greater than 2 it can be understood as a gradient flow of the total variation energy in D^{-1},the dual space of D^1_0, which is a completion of the space of compactly supported smooth function in the Dirichlet norm. However, if n is less than or equal to 2, the space D^{-1} imposed average free condition at least formally. Since we are interested in motion of characteristic functions, we extend the notion of a solution in a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a kind of duality argument.
We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout evolution.
We prove that all balls are calibrable. However, unlike in the second order problem, the outside a ball is not calibralbe if and only if n=2. If n is not 2, all annuli are calibrable, while in the case n=2, it is not calibrable if an annulus is too thick. We give a whole evolution of the characteristic function of a ball. We even study evolution of several piecewise constant radial functions. This is a joint work of Hirotoshi Kuroda (Hokkaido University) and Michał Łasica (Polish Academy of Sciences/University of Tokyo).
Zoom Information
https://zoom.us/j/95704298808?pwd=NGs0WEtlV2RuL0tTNlVwaE52RTY2QT09
Conference number：957 0429 8808
Passwords：839373
