Abstract:  We consider the stochastic damped nonlinear wave equation in two dimensions. We show that, in the entire non-singular regime, the law of the solution remains absolutely continuous with respect to the law of the solution to the linear stochastic damped wave equation. Whilst such results are typically well-known for stochastic parabolic equations, the associated techniques fail in the hyperbolic setting at low-regularity. Our argument extends to the stochastic setting recent ideas in the study of the quasi-invariance of Gaussian measures under the flow of Hamiltonian PDEs.

This talk is based on joint work with L. Tolomeo (University of Edinburgh).

Abstract: Since a seminal paper by Jordan, Otto, and Kinderlehrer (98'), it is now well known that some evolution PDEs, such as diffusion and advection PDEs, can be interpreted as gradient flows with respect to the Wasserstein distance. Since then, there have been ongoing efforts to integrate various evolutionary processes into this framework. In this talk, I will introduce the Moran process and its high popuation limit the Kimura equation, and explain how they are related to Wasserstein gradient flows. Indeed, the degeneracy of the diffusion at the boundaries leads to the study of a conditioned version of the dynamics, that can be seen as a Wasserstein gradient flow, with degenerate underlying geometry, involving the Shahshahani metric. Finally, we will see that the dissipation induced by the hidden gradient flow in the continuous setting is a good approximation of the dissipation in the discrete setting in its high population limit.

Abstract: In this presentation, we give a sufficient condition to guarantee the existence of a smooth solution of the d-dimensional Navier-Stokes equation with nice decreasing properties at infinity for dimension greater than or equal to 3. In this way, we prove the existence of smooth physically reasonable solutions to the Navier-Stokes problem. Additionally, we show the existence of a smooth curve of entire vector fields of order 2 that extends the solution to the complex domain for positive time.

Abstract: The Monge-Ampère type equations over bounded convex domains arise in a host of geometric applications. In this talk, we focus on the Dirichlet problem for a class of Monge-Ampère type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate a series of results regarding existence, uniqueness, interior regularity, and boundary regularity of the viscosity solutions to the problem.

Abstract: The detailed balance is a property of macroscopic systems that are obtained from an underlying time-reversible microscopic model. It states that each elementary process (for instance each chemical reaction) is in equilibrium with its reverse process. Even if, at the fundamental level, we expect chemical reactions to satisfy the so-called detailed balance condition, biochemical systems are often modeled by systems of equations for which detailed balance fails. This can be justified if the system is in contact with "reservoirs" that are out of equilibrium, as is usually the case in biological systems. In this talk, I will discuss the relation between the detailed balance property and the property of adaptation of chemical systems. In particular, a system satisfies the adaptation property if it responds to gradients instead of absolute values of signals. I will show that, unless a factorization assumption on the conservation laws holds, the property of adaptation cannot hold in a stable manner in closed systems (i.e., systems that satisfy the detailed balance property and do not exchange substances with the environment).

Abstract: We present improved regularity results for solutions to a nonlocal dead-core problem at branching points. Our approach, which does not rely on the maximum principle, introduces a new strategy for analyzing two-phase problems within the local framework, an area that remains largely unexplored.

This is a joint work with D. dos Prazeres (UFS, Brazil) and J.M. Urbano (KAUST, KSA & University of Coimbra, Portugal).

Abstract: Coagulation and fragmentation equations are mathematical models that describe the processes of particle aggregation (coagulation) and division (fragmentation) within a population. These equations, utilized in fields such as chemistry, physics, and biology, are crucial for understanding the dynamics of particle systems. Coagulation equations depict how particles merge to form larger entities, while fragmentation equations illustrate the breakdown of larger particles into smaller ones.

The Becker-Döring equations, introduced by Becker and Döring (1935), serve as fundamental frameworks for describing first-order phase transitions in various chemical and physical systems, such as vapor condensation and lipid aggregation. Over the years, significant attention has been devoted to studying this model in the nonlinear case in contrast to the linear situation, such as the work by Penrose (1989) where metastable states were studied, or Cañizo & Lods (2013), studying the exponential convergence of the solution only in the subcritical case. For this reason, we focus on the linear Becker-Döring equations, defined by:

d/dt ci(t) = W_{i−1} − W_ i , for all i ≥ 2

being c1 constant for all t∈ℝ, where the fluxes W_i depend on the coagulation and fragmentation coefficients a_i and b_i, respectively, following the expression W_i = a_i c_1 c_i − b_{i+1} c_{i+1}.

Our aim is to examine the long-term behaviour of solutions to the equilibria, providing a more accurate estimation of the convergence rate than that shown by Kreer (1993). To achieve this, we employ spectral theory and entropy methods, using the free energy provided also by Kreer (1993).