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Register a new userRegularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods
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In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.
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We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space $mathcal S$ and in the space ${mathcal S}_omega$ as introduced by Bjorck for weight functions $omega$. Several examples are discussed in this new setting.
We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.
Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.
We prove global existence, uniqueness and stability of entropy solutions with $L^2cap L^infty$ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a unique continuous manner to all $L^2$ initial data. These weak solutions are found to satisfy one sided Holder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.
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