Mathematical Sciences: Nonlinear Evolution Equations

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Abstract Ponce The project is devoted to research into various aspect of nonlinear evolution equations. They are at the intersection of partial differential equations, classical harmonic analysis and mathematical physics. The principal areas include the study of several dispersive systems arising as approximate models in wave propagation problems. The emphasis is on the development of arguments and techniques, mainly from Fourier analysis, which provide a better description of the behavior of the solutions of such models.The second part of the project is devoted to the study of problems related to viscous flows. They include the long time behavior of solutions to the 2-D density dependent Navier-Stokes equations. Also the system for compressible ideal fluids under minimal regularity of the data is studied. In concrete problems the validity of these models as a good approximation of the physical phenomenon depends on specific properties of their solutions, mainly those describing their long time behavior. Thus, the qualitative features of solutions obtained through this research provide a benchmark against which numerical approximation schemes can be tested. This applied analysis of nonlinear partial differential equations is an equal partner with their numerical study and the modelling of the physical phenomena, all of which interact to give a full mathematical picture.