TSUHARA Shun Researcher
Research Activities
My research interest is partial differential equations(PDEs) derived from various physical phenomena. My aim is to reveal properties of the solution to PDEs, such as well-posedness, global behavior in time and so on. I am also interested in the method of mathematical analysis in functional analysis, real analysis and harmonic analysis. Recently, I am studying the nonlinear Schrödinger equation(NLS) descriving the model of nonlinear optics. NLS is one of the nonlinear dispersive equations that has linear dispersion effects, in which waves scatters, and nonlinear interactions, with which waves gather, and it is well known that behaviors of the solutions to NLS are various. For the critical problem where dispersion and nonlinearity are balanced, there is a threshold of initial data, which divides the global existence and blowing up in finite time to the solutions. I aim to determine such a threshold for the system of criticla NLS, or the single critical NLS in the half space. For the system of NLS, I focus on the Hamiltonian structures and treat the general nonlinearities including previous nonlinear terms. For the problem on the half space, I am treating the nonlinear boundary condition, however, there are still few previous works for the NLS with the nonlinear boundary term. I am constructing the new functional inequalities in the half space by many methods of mathematical analysis, and applying that to NLS step by step.
Papers
- S. Tsuhara, Global well-posedness for the Sobolev critical nonlinear Schrödinger system with general nonlinear terms, Proceedings of the conference “Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities.”, accepted for publication.
- T. Ogawa, S. Tsuhara, Wellposedness for the nonlinearSchrödinger equation with the nonlin ear boundary condition in low dimensional half spaces, Adv. Stud. Pure Math., accepted for publication.
- T. Ogawa, T. Sato, S. Tsuhara, The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane, NoDEA Nonlinear Differ. Equ. Appl. 31 (2024), No. 59, 22pp.
- T. Ogawa, S. Tsuhara, Global well-posedness for the Sobolev critical nonlinear Schrödinger system in four space dimensions, J. Math. Anal. Appl. 524 (2023), No. 127052, 27 pp.
Contact
tsuhara (at) math.sci.hokudai.ac.jp