In the present talk, we discuss the properties and asymptotic stability of spherically symmetric stationary solutions to the heat-conductive compressible Navier-Stokes equations in the exterior domain of a unit ball with out-flowing boundary condition. The existence of the stationary solution is proved by Professor Akitaka Matsumura and Itsuko Hashimoto. We derive several properties of the stationary solution, especially convergence rates towards the far field. By making use of these rates, we derive a-priori estimate of the perturbation from the stationary solution in the suitable weighted Sobolev space. It shows the asymptotic stability of the stationary solution. In this derivation, the relative energy form plays an essential role. We also discuss several recent results on the isentropic flow with out/in-flowing boundary condition.

六角格子上の離散シュレディンガー方程式はグラフェン上の電子を強束縛近似で記述するモデル方程式として知られている．本講演ではポテンシャルがない場合の自由発展における$L^1 \to L^\infty$評価を各運動量について局所的に得られ，最も遅い時間減衰が$|t|^{-2/3}$のオーダーであることを示す．証明は運動量空間における振動積分で得られるが，位相関数が特異になるディラック点ではVarchenkoの定理のような一般論が適用できないため初等的な積分計算によって評価する．本講演はYounghun Hong氏 (Chung-Ang University), Changhun Yang氏 (Chungbuk National University)との共同研究に基づく．