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Browsing by Subject "Sobolev space"

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  • Laarne, Petri (2021)
    The nonlinear Schrödinger equation is a partial differential equation with applications in optics and plasma physics. It models the propagation of waves in presence of dispersion. In this thesis, we will present the solution theory of the equation on a circle, following Jean Bourgain’s work in the 1990s. The same techniques can be applied in higher dimensions and with other similar equations. The NLS equation can be solved in the general framework of evolution equations using a fixed-point method. This method yields well-posedness and growth bounds both in the usual L^2 space and certain fractional-order Sobolev spaces. The difficult part is achieving good enough bounds on the nonlinear term. These so-called Strichartz estimates involve precise Fourier analysis in the form of dyadic decompositions and multiplier estimates. Before delving into the solution theory, we will present the required analytical tools, chiefly related to the Fourier transform. This chapter also describes the complete solution theory of the linear equation and illustrates differences between unbounded and periodic domains. Additionally, we develop an invariant measure for the equation. Invariant measures are relevant in statistical physics as they lead to useful averaging properties. We prove that the Gibbs measure related to the equation is invariant. This measure is based on a Gaussian measure on the relevant function space, the construction and properties of which we briefly explain.