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Well-posedness and qualitative properties for nonlinear PDEs


In this project we will address well-posedness and qualitative properties of nonlinear partial differential equations (PDEs) distributed in the following groups: elliptic equations and systems; fluid dynamics equations; and hyperbolic and dispersive equations. We will analyze how initial and boundary conditions, properties of domains, nonlinearities, potentials and forcing terms, presenting certain types of symmetries, anisotropies and singularities, influence the behavior of solutions. We will investigate issues such as existence, uniqueness, continuous dependence on parameters and data, symmetries, singularities, self-similarity, decay, stability, and asymptotic behavior. The general approach in the project is to analyze the equations in functional spaces that allow data, coefficients and solutions with the desired properties, such as spaces of Radon measures, $L^{\infty}$ with homogeneous weight (and sum of translations of those), Morrey spaces, Besov, modulation spaces, Fourier-Besov, Herz-type spaces, weak-Herz, Besov-Herz, Besov-Morrey, Fourier-Besov-Morrey, among others. The handling of the integral operators linked to the notion of solution requires interpolation techniques, product, convolution and commutator operator estimates, characterizations of preduals and block spaces, estimates for volume-preserving diffeomorphisms, among other ingredients of harmonic analysis. (AU)

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