Title: Functional Equations In Rings

Abstract

Title: Interpolation spaces and Hölderian mappings, applications to Quasilinear P.D.E.s

Abstract: As in the work of Tartar [1], we develop here some new results on nonlinear interpolation of α-Hölderian mappings between normed spaces, by studying the action of the mappings on K-functionals and between interpolation spaces with logarithm functions. We apply these results to obtain some regularity results on the gradient of the solutions to quasilinear equations of the form
– div (â(∇ u ))+V(u)=f,
where V is a nonlinear potential and f belongs to non-standard spaces like Lorentz-Zygmund spaces. We show several results; for instance, that the mapping T: Tf=∇u is locally or globally α-Hölderian under suitable values of α and appropriate hypotheses on V and a ̂.

The talk is based on the paper [2].
References
[1] Tartar L., Interpolation non linéaire et régularité, J. Funct. Anal. }, 9 (1972), 469--489. [2] Ahmed, I. Fiorenza, A.; Formica, M. R.; Gogatishvili, A.; El Hamidi, A.; Rakotoson, J. M., Quasilinear PDEs, interpolation spaces and Hölderian mappings, Anal. Math. 49 (2023), no. 4, 895--950.

Title: Limits of sequence of Tensor Product Operators connected to the Walsh-Paley system

Abstract

Title: Applications of Lie Groups to Differential Equations

Abstract: During the latter half of the nineteenth century, the distinguished Norwegian mathematician Marius Sophus Lie (1842-1899) developed the theory of continuous groups and applied it to the study of geometry and differential equations. His investigations led to one of the major branches of 20th-century mathematics, the theory of Lie groups and Lie algebras. This technique systematically connects and broadens the well-known ad hoc methods to construct closed-form solutions of differential equations, particularly for nonlinear differential equations. In this talk we present Lie’s theory in brief and its applications to differential equations.

Title: Cauchy problem for linear and nonlinear nonlocal Ginzburg-Landau equations

Abstract: In this study, the Cauchy problem for linear and nonlinear nonlocal Ginzburg-Landau equations is mentioned. The equations include variable coefficients with convolution terms. Here, assuming enough smoothness on the initial data and the growth conditions on coefficients, we obtain first the separability properties of the corresponding linear Ginzburg-Landau equation. Then based on the maximal regularity properties of linear problem and perturbation techniques the existence, uniqueness of local and global solution, L^{p}-regularity, and blow up properties to solution of nonlinear Ginzburg-Landau equations are established.