Author Archives: rmartinez

Derivations of wave kinetic equations in 1D

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The HADES seminar on Tuesday, September 8th, will be at 3:30pm in Room 736.

Speaker: Katja Vassilev

Abstract: Wave kinetic equations give an effective description of dispersive systems. In this talk, we look at various 1D dispersive systems and show how a diagrammatic expansion of the solution to the dispersive system gives a framework for these derivations. In particular, we discuss the features of these dispersive systems which restrict the timescale to which these derivations can currently be done. Such features include the dimension of the system, resonances in the nonlinearity, and lack of invariances.

Existence of weak solutions to a model of the geodynamo

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The HADES seminar on Tuesday, March 10th, will be at 3:30pm in Room 740.

Speaker: Tom Schang

Abstract: In this talk, I will discuss a model of the earth’s magnetic field that has previously been simulated numerically, but has not been shown to be well-posed. This model couples solid physics for the electrically conducting inner core with magnetohydrodynamic (MHD) equations in the liquid outer core, as well as the magnetic field outside of the core, which is taken to be non-conducting. I will define and prove existence of Leray-Hopf-type weak solutions for this system. Particular challenges include the transmission conditions of the magnetic field coming from non-constant physical parameters and extending the magnetic field to a non-conducting exterior. To address these problems, we must carefully define a function space and prove appropriate embeddings.

Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

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The HADES seminar on Tuesday, February 24th, will be at 3:30pm in Room 740.

Speaker: Hongjing Huang

Abstract:

We consider one-dimensional scalar quasilinear Klein–Gordon equations with general nonlinearities, on both $\mathbb R$ and $\mathbb T$.
By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions.
Our main result asserts  that solutions with small initial data of size $\epsilon$ persist on the improved cubic timescale $|t| \lesssim \epsilon^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case 
of $\mathbb R$, we are further able 
to use dispersion in order to extend the lifespan to $\epsilon^{-4}$. This generalizes earlier results 
obtained by Delort, \cite{Delort1997_KG1D}  in the semilinear case.
This joint work with Mihaela Ifrim and Daniel Tataru.