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We consider focusing nonlinear Schr\"odinger equations, which are mass supercritical and energy subcritical. For these equations, given initial data which has mass and energy below ground state initial data, a classification between blow up and global existence is known. However for initial data above the ground state, behavior is only known for special cases. In this talk we discuss the impact of angular momentum on long time behavior of solutions and introduce some new global and blow up solutions.
New global and blowup solutions for nonlinear Schrodinger equations above the ground state
Fri, Oct. 18 4pm (MATH 220)
Siddhant Agrawal (University of Colorado Boulder)
X
We consider the 2D incompressible Euler equation on a bounded simply connected domain. We give sufficient conditions on the domain so that for all bounded initial vorticity, the weak solutions are unique. Our sufficient conditions allow us to prove uniqueness for a large subclass of domains and convex domains. Previously uniqueness for general bounded initial vorticity was only known for domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below the regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional. This is joint work with Andrea Nahmod.
Uniqueness of the 2D Euler equation on rough domains
Tue, Nov. 12 1:25pm (Math 3…
Gregory Berkolaiko (Texas A&M)
X
In this overview talk we will explore a variational approach to the problem of Spectral Minimal Partitions (SMPs). The problem is to partition a domain or a manifold into k subdomains so that the first Dirichlet eigenvalue on each subdomain is as small as possible. We expand the problem to consider Spectral Critical Partitions (partitions where the max of the Dirichlet eigenvalues is experiencing a critical point) and show that a locally minimal bipartite partition is automatically globally minimal.
Extensions of this result to non-bipartite partitions, as well as its connections to counting nodal domains of the eigenfunctions and to a two-sided Dirichlet-to-Neumann map defined on the partition boundaries, will also be discussed.
The talk is based on joint papers with Yaiza Canzani, Graham Cox, Bernard Helffer, Peter Kuchment, Jeremy Marzuola, Uzy Smilansky and Mikael Sundqvist.
Spectral minimal partitions: local vs global minimality
New global and blowup solutions for nonlinear Schrodinger equations above the ground state