arXiv subjects
Publications and source records attributed to Masashi Misawa.
In this paper we prove the existence of a weak solution to a doubly nonlinear parabolic fractional $p$-Laplacian equation, which has general doubly non-linearlity including not only the Sobolev subcritical/critical/supercritical cases but also the slow/fast diffusion ones. Our proof reveals the weak convergence method for the doubly nonlinear fractional $p$-Laplace operator.
We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case p=2. In this article we establish a priori estimates and regularity results for the $p$-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.
In this paper, we study a doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow. In the special case p=2 our theory includes the classical Yamabe flow on a bounded domain in Euclidean space. Our main aim is to prove the global existence of the p-Sobolev flow together with its qualitative properties.
Asymptotic profile for diffusion wave terms of solutions to the compressible Navier-Stokes-Korteweg system is studied on $R^2$. The diffusion wave with time decay estimate is studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002) and Kobayashi and Tsuda (2018) for the compressible Navier-Stokes system and the compressible Navier-Stokes-Korteweg system. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space-time $L^2$ of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by $L^2$ on space, a decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz's energy estimate, and the Fefferman-Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.