This dissertation consists of three parts. In the

This dissertation consists of three parts. In the

The subject of the present thesis is the Chern-Simons-Schrödinger system, which is a gauge-covariant Schrödinger system in two spatial dimensions with a long-range electromagnetic field. The present thesis studies two aspects of the system: that of well-posedness and that of the long-time behaviour. The first main result of the thesis concerns the large-data well-posedness of the initial-value problem for the Chern-Simons-Schrödinger system. We impose the Coulomb gauge to remove the gauge-invariance, in order to obtain a well-defined initial-value problem. We prove that, in the Coulomb gauge, the Chern-Simons-Schrödinger system is locally well-posed in the Sobolev spaces $H^s$ for $s\ge 1$, and that the solution map satisfies a weak Lipschitz continuity estimate. The main technical difficulty is the presence of a derivative nonlinearity, which rules out the naive iteration scheme for proving well-posedness. The key idea is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and to exploit the dispersive properties of the resulting paradifferential-type principal operator, in particular frequency-localised Strichartz estimates, using adaptations of the $U^p$ and $V^p$ spaces introduced by Koch and Tataru in other contexts. The other main result of the thesis characterises the large-time behaviour in the case where the interaction potential is the defocusing cubic term. We prove that the solution to the Chern-Simons-Schrödinger system in the Coulomb gauge, starting from a localised finite-energy initial datum, will scatter to a free Schrödinger wave at large times. The two crucial ingredients here are the discovery of a new conserved quantity, that of a pseudo-conformal energy, and the cubic null structure discovered by Oh and Pusateri, which reveals a subtle cancellation in the long-range electromagnetic effects. By exploiting pseudo-conformal symmetry, we also prove the existence of wave operators for the Chern-Simons-Schrödinger system in the Coulomb gauge: given a localised finite-energy final state, there exists a unique solution which scatters to that prescribed state.

We consider three different phenomena governing the fluid flow in the presence of charged particles: electroconvection in fluids, electroconvection in porous media, and electrodiffusion. Electroconvecton in fluids is mathematically represented by a nonlinear drift-diffusion partial differential equation describing the time evolution of a surface charge density in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to Navier-Stokes equations forced nonlinearly by the electrical forces due to the presence of the charge density. The resulting model is reminiscent of the quasi-geostrophic equation, where the main difference resides in the dependence of the drift velocity on the charge density. When the fluid flows through a porous medium, the velocity and the electrical forces are related according to Darcy’s law, which yields a challenging doubly nonlinear and doubly nonlocal model describing electroconvection in porous media. A different type of particle-fluid interaction, called electrodiffusion, is also considered. This latter phenomenon is described by nonlinearly advected and nonlinearly forced continuity equations tracking the time evolution of the concentrations of many ionic species having different valences and diffusivities and interacting with an incompressible fluid. This work is based on [1, 2, 3] and addresses the global well-posedness, long-time dynamics, and other features associated with the aforementioned three models. REFERENCES:[1] E. Abdo, M. Ignatova, Long time dynamics of a model of electroconvection, Trans. Amer. Math. Soc. 374 (2021), 5849–5875. [2] E. Abdo, M. Ignatova, Long Time Finite Dimensionality in Charged Fluids, Nonlinearity 34 (9) (2021), 6173–6209. [3] E. Abdo, M. Ignatova, On Electroconvection in Porous Media, to appear in Indiana University Mathematics Journal (2023). / Mathematics

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