In this seminar, I will discuss the local well-posedness (LWP) of the initial value problem for the Hirota-Satsuma system - a coupled system of two KdV-type equations. Physically, this system models nonlinear waves interactions of long waves with different dispersion relations, with applications in shallow water and stratified fluids. Since there are two equations in this system, we can suppose the initial data have different regularities, i.e., $(u_0, v_0) \in H^k (\mathbb{R}) \times H^s(\mathbb{R})$. Previous LWP results in the literature only suppose $k= s$ or $s = k + 1$. Using frequency-restricted estimates and the concept of integrated-by-parts strong solution, we were able to prove LWP in the general case $H^k (\mathbb{R}) \times H^s(\mathbb{R})$. This LWP is sharp except for only one obstruction which we are still investigating.

This is a joint work with Simão Correia and Jorge Drumond.

The replicator equation, originally used in evolutionary game theory, has been widely applied in evolutionary biology and ecology to model the complex dynamics of multi-type interacting species, such as multi-species ecological ensembles or multi-strain microbial pathogens. In this seminar we explore aspects of the competitive dynamics encoded in the replicator equation, paying special attention to invader-driven systems, in which all species are proactive. We focus on relating properties of the fitness matrix to the quality of species dynamics and to the diversity of these systems at equilibrium, and we find a mechanism for the selection of the final set of surviving species.

In this talk, we introduce a parametric bilinear fractional integral operator and obtain uniform bounds via Marcinkiewicz interpolation. This operator is motivated by a reformulation of Euler-Riesz systems, where the nonlocal term in the momentum equation is expressed as the divergence of a tensorial integral operator. This reformulation is relevant as it leads to an a priori higher integrability estimate for the density of finite-energy solutions. The uniform bounds obtained for the parametric fractional integrals then yield bounds for the tensorial integral operator. This is based on joint work with L. Grafakos and A. E. Tzavaras.

In this talk we consider differential operators of the form $\text{div}(A \, \text{grad})$ where the coefficients of $A$ are measurable and bounded functions and the eigenvalues of $A$ are positive and uniformly bounded from $0$. A solution on a domain $D \subset \mathbb{R}^d$ is a function that satisfies

\[ \operatorname{div}(A \, \operatorname{grad}) u = 0 \text{ on } D. \]

We examine boundary behavior of the gradient of solutions to such differential operators. The classical case, where the differential operator in question is the Laplacian, is well understood for $C^2$ domains due to Havin and Mozolyako [1] and later also for Lipschitz domains due to Müller and Riegler [2]. This talk mentions techniques to obtain estimates on the radial variation of positve harmonic functions and also gives an outlook of how to adapt the existing techniques to general uniformly elliptic divergence-form operators or solutions on more complicated domains.

1. Pavel A. Mozolyako and Viktor P. Khavin. Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra i Analiz 28.3 (2016), pp. 67–110. issn: 0234-0852. doi: 10.1090/spmj/1454. url: https://doi.org/10.1090/spmj/1454.
2. Paul F. X. Müller and Katharina Riegler. Radial variation of Bloch functions on the unit ball of Rd. Ark. Mat. 58.1 (2020), pp. 161–178. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2020.v58.n1.a10. url: https://doi.org/10.4310/ARKIV.2020.v58.n1.a10.