arXiv subjects

Katie Marsden

Publications and source records attributed to Katie Marsden.

4 recordsLinked to original sources

arXiv2026

A splitting scheme for the wave maps equation at low regularity

We prove convergence of a filtered Lie splitting scheme for the wave maps equation with low regularity initial data in dimension 3. The convergence analysis is performed in discrete Bourgain spaces, as has proved fruitful for the low regularity analysis of the equation in the continuous setting. An important difficulty here is that the analysis of wave maps at low regularity requires the use of the null structure of the system, this structure thus has to be preserved at the discrete level to get an effective stable low regularity scheme. Since the null structure involves time derivatives, the scheme has to be designed carefully. The presence of time derivatives in the nonlinearity then constitutes the most significant source of numerical error. Nonetheless, we are able to prove convergence of the scheme for all subcritical initial data in $H^s$, $s>d/2$.

math.NA↗

arXiv2026

The Hamiltonian formulation of continuum Calogero-Moser models

Recent well-posedness results have identified the Hardy space $L^2_+$ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form on this phase space and so are able to realize these models as Hamiltonian systems. Moreover, we demonstrate that previously identified conserved quantities are mutually commuting, reinforcing the notion that these models are completely integrable. We further illustrate the utility of these structures by using them to give a new proof of global well-posedness in the critical space $L^2_+$, under the necessary mass restriction in the focusing case. Our work also brings to light several unforeseen connections: (i) the threshold for well-posedness coincides with that for the nondegeneracy of the symplectic form; (ii) this threshold is connected through Carleman's inequality to the isoperimetric problem in the plane; (iii) the transition from the line to the torus gives rise to a modified dynamical equation.

math.AP↗

arXiv2024

Global Solutions to the 3D Half-Wave Maps Equation with Angular Regularity

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions $n\geq4$, global wellposedness for data which is small in the critical Besov space $\dot{B}^{n/2}_{2,1}$ is known since the works of Krieger, Sire and Kiesenhofer [13,9]. There is a major obstruction in extending these results to three dimensions due to the loss of the crucial $L^2_tL^\infty_x$ Strichartz estimate. In this work, we make progress on this case by proving that the equation is "weakly" globally well-posed (in the sense of Tao [28]) for initial data which is not only small in $\dot{B}^{3/2}_{2,1}$ but also possesses some angular regularity and weighted decay of derivatives. We use Sterbenz's improved Strichartz estimates in conjunction with certain commuting vector fields to develop trilinear estimates in weighted Strichartz spaces which avoid the use of the $L^2_tL^\infty_x$ endpoint.

math.AP↗

arXiv2022

Almost Sure Scattering of the Energy Critical NLS in $d>6$

We study the energy-critical nonlinear Schr\"{o}dinger equation with randomised initial data in dimensions $d>6$. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $H^s(\mathbb{R}^d)$ whenever $s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\}$. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results of Spitz in dimension 4. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.

math.AP↗