TAILIEUCHUNG - Đề tài " Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature "

For the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke’s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen. Introduction In this paper we study the asymptotic analysis, as the parameter ε goes to zero, of the complex-valued parabolic Ginzburg-Landau equation for functions uε : | Annals of Mathematics Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature By F. Bethuel G. Orlandi and D. Smets Annals of Mathematics 163 2006 37 163 Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature By F. Bethuel G. Orlandi and D. Smets Abstract For the complex parabolic Ginzburg-Landau equation we prove that asymptotically vorticity evolves according to motion by mean curvature in Brakke s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases we also prove convergence to enhanced motion in the sense of Ilmanen. Introduction In this paper we study the asymptotic analysis as the parameter E goes to zero of the complex-valued parabolic Ginzburg-Landau equation for functions u RN X R C in space dimension N 3 ldu- - us -2ue 1 - u 2 on Rn X 0 to ơt E2 u x 0 u0 x for x E Rn. PGL This corresponds to the heat-flow for the Ginzburg-Landau energy E u Ị e u dx Ị u V u dx for u Rn C where V denotes the nonconvex potential V u . This energy plays an important role in physics and has been studied extensively from the mathematical point of view in the last decades. It is well known that PGL is well-posed for initial data in Hloc with finite Ginzburg-Landau energy E u0 . Moreover we have the energy identity I E ue T2 due dt 2 x t dx dt S u - T1 V 0 T1 T2 . This work was partially supported by European RTN Grant HPRN-CT-2002-00274 Front Singularities . 38 F. BETHUEL G. ORLANDI AND D. SMETS We assume that the initial condition u0 verifies the bound natural in this context H0 u0 Mo log 4 where M0 is a fixed positive constant. Therefore in view of I we have II E u - T E u0 Mo loge for all T 0. The main emphasis of this paper is placed on the asymptotic limits of the Radon measures a defined on RN X R by a x t ogy dxdt and of their time slices a defined on RN X t by t e u x t at x 77-- i dx log e so that a a dt. In view of assumption H0 and II we may assume up to a subsequence en

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