For the Schrodinger map equation $${u_t = u \times \triangle u \, {\rm in} \, \mathbb{R}^{2+1}}$$
 , with values in S
 2, we prove for any $${\nu > 1}$$
 the existence of equivariant finite time blow up solutions of the form $${u(x, t) = \phi(\lambda(t) x) + \zeta(x, t)}$$
 , where $${\phi}$$
 is a lowest energy steady state, $${\lambda(t) = t^{-1/2-\nu}}$$
 and $${\zeta(t)}$$
 is arbitrary small in $${\dot H^1 \cap \dot H^2}$$
 .

Blow Up Dynamics for Equivariant Critical Schrödinger Maps

In this article, we establish the unconditional uniqueness of solutions to an infinite radial Chern–Simons–Schrodinger (IRCSS) hierarchy in two spatial dimensions. The IRCSS hierarchy is a system of infinitely many coupled PDEs that describes the limiting Chern–Simons–Schrodinger dynamics of infinitely many interacting anyons. The anyons are two-dimensional objects that interact through a self-generated field. Due to the interactions with the self-generated field, the IRCSS hierarchy is a system of nonlinear PDEs, which distinguishes it from the linear infinite hierarchies studied previously. Factorized solutions of the IRCSS hierarchy are determined by solutions of the Chern–Simons–Schrodinger system. Our result therefore implies the unconditional uniqueness of solutions to the radial Chern–Simons–Schrodinger system as well.

/pdf/on-the-unconditional-uniqueness-of-solutions-to-the-infinite-3grni85ei4.pdf

On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy

product on C n R developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H p flag , as well as from H p flag to L p , for 0< p 1. We also characterize the dual spaces of H 1 flag and H p flag , and establish a Calderon‐Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H p flag. In particular, this recovers some L p results of Muller, Ricci, and Stein (but not their sharp versions) by interpolating between those for H p flag and L 2 .

/pdf/flag-hardy-spaces-and-marcinkiewicz-multipliers-on-the-3gsg001obr.pdf

Flag hardy spaces and marcinkiewicz multipliers on the heisenberg group

Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the lack of a two parameter group of automorphic dilations on H^{n}. This lack of dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H^{n} when 0<p\leq1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product Hardy spaces H_{product}^{p} on C^{n}\timesR. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H_{flag}^{p}, as well as from H_{flag}^{p} to L^{p}, for 0<p\leq1. We characterize the dual spaces of H_{flag}^{1} and H_{flag}^{p}, and establish a Calder\'on-Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H_{flag}^{p}. In particular, this recovers the L^{p} results by interpolating between those for H_{flag}^{p} and L^{2} (but regularity sharpness is lost).

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version

We consider the energy-critical half-wave maps equation $$\begin{aligned} \partial _t {\mathbf {u}}+ {\mathbf {u}}\wedge |\nabla | {\mathbf {u}}= 0 \end{aligned}$$
 for $${\mathbf {u}}: [0,T) \times {\mathbb {R}}\rightarrow {\mathbb {S}}^2$$
 . We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $${\mathbb {S}}^2$$
 . In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $${\mathbb {S}}^2$$
 applied to half-harmonic maps from $${\mathbb {R}}$$
 to $${\mathbb {S}}^2$$
 . Complementing our classification result, we carry out a detailed analysis of the linearized operator L around half-harmonic maps $${\mathbf {Q}}$$
 with arbitrary degree $$m \geqslant 1$$
 and consisting of m identical Blaschke factors. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $${\mathbf {Q}}$$
 . Moreover, we give a full description of the spectrum of L by finding all its $$L^2$$
 -eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for L and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $${\mathbb {R}}$$
 to the unit circle $${\mathbb {S}}$$
 . Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.

On energy-critical half-wave maps into $${\mathbb {S}}^2$$ S 2

We consider equivariant solutions for the Schrodinger map problem from ℝ2+1 to S2 with energy less than 4π and show that they are global in time and scatter.

/pdf/equivariant-schrodinger-maps-in-two-spatial-dimensions-1a0s712jxp.pdf

Adrian Mihai Ionescu

Papers

Equivariant schrödinger maps in two spatial dimensions