One Day Workshop at NCKU

DATE2023-03-08 16:10-17:00

PLACE數學系館 3F會議室

SPEAKERProfessor Kenji Nakanish(Kyoto University and RIMS

TITLEOptimal Wellposednesss for the Hartree Equation in the Schatten Class

ABSTRACT Singularity formation and Cauchy Problem for Some Nonlinear PDEs
Organizers: 方永富,史習偉,郭鴻文,關汝琳
Goal: We arrange a series of lectures at introductory level and hope to attract students to appreciate some merits of the PDE topic

This talk is based on joint work with Sonae Hadama (Kyoto). We consider a mean-field approximation model to describe the dynamics of many fermion particles, which is a large system of Schrodinger equations with the Hartree interactions among orthogonal functions. Lewin and Sabin ('14-15) proved stability of some stationary solutions with translation invariance, which have physical relevance such as for ideal Fermi gas.
A key tool in their analysis was the Strichartz estimate for orthogonal functions, where the total mass (L^2) is not summable but merely p-th power summable with respect to the number of particles for some p>1.
When the equation is written in the operator form for the orthogonal projections, such solutions belong to the Schatten-p class, while the case of finite total mass corresponds to the trace class.
Our question is how much p is allowed for the Cauchy problem to be well-posed for initial data in the Schatten-p class. We give an almost complete answer in the case of the Riesz potential, namely the exact range of Schatten exponent, depending on the power of potential and the spatial dimensions.
To prove the positive side, namely the wellposedness, we introduce a norm for propagators corresponding to the best constants of the Strichartz estimate in the Schatten class, and a Schatten version of the Christ-Kiselev lemma for the Duhamel integral. To prove the negative side, namely the illposedness, we take sequences of initial data, known for the optimality of the Schatten Strichartz estimate, and show discontinuity of the flow map by constructing approximate solutions.