For en løsning av en partielle differensialligning er det ofte av interesse å vite hvor (dvs. på hvilket område) løsningen er analytisk, fordi dette kan gi innsikt i de underliggende fysiske prosessene. I dette prosjektet studerer vi ligningene for magnet-hydrodynamikk (MHD) samt flere andre modeller med opphav i studiet av bølgebevegelser i fluidmekanikk.
MHD-ligningene er av grunnleggende betydning i studiet av kjernefysisk fusjon. I bestrebelsene på å forstå plasma-dynamikk, spiller forståelsen av anomal transport drevet av turbulens i MHD en viktig rolle. En bedre forståelse av geometrien til løsningene er viktig for å kunne konstruere mer effektive fusjons-reaktorer, inkludert ITER og neste generasjon reaktorer.
The proposed research is aimed at extending the use of function-theoretic methods in the study of nonlinear partial differential equations with a view towards applications to important problems in mathematical physics. Information about the domain of anal yticity of a solution of a PDE can be used to gain a quantitative understanding of the structure of the equation, and to obtain insight into underlying physical processes. The study of real-analytic solutions to nonlinear PDE has developed over the last t wo decades, and analytic function spaces have become popular tools for the study of a variety of questions connected with nonlinear evolutionary PDE. In particular, the use of Gevrey-type spaces has given rise to a number of important results in the study of long time dynamics of dissipative equation, such as estimating the asymptotic degrees of freedom, and approximating the global attractors. Here, we want to expand the range of these techniques and applications in the context of dissipative and dispers ive equations.
The equations under study include the three-dimensional magneto-hydrodynamics system, and the nonlinear Maxwell-Dirac system. In view will also be nonlocal model equations for fluid flow, such as the Benjamin-Ono equation. In the context of magnetohydrodynamics, the proposed research will focus on the study of possible singularity formation, and scenarios to prevent such a collapse. The Maxwell-Dirac system appears in the study of quantum electrodynamics, and has recently been studied in the context of low-regularity solutions. However, results about the domain of analyticity of spatially-analytic solutions appear to be lacking. Regarding nonlocal evolution equations, our goal will be to obtain a theory of solutions in classes of analytic functions which parallels the existing theory for third-and higher order differential equations. We will also exploit analyticity of solutions in the study of the convergence of numerical approximation schemes.