학과 세미나 및 콜로퀴엄




2022-08
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2022-09
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In this talk, we present a short history of Lp theories for (stochastic) partial differential equations. In particular, we introduce recent developments handling degenerate equations in weighted Sobolev spaces. It is well known that there exist probabilistic representations of solutions to second order (stochastic) partial equations, which enables us to use many interesting probabilistic theories to investigate solutions. Recently, by applying probabilistic tools, we obtain interesting new type weighted estimates to second order degenerate (stochastic) partial differential equations.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-09-16 17:31:07
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations \begin{equation} \mathrm{St} \partial_t F + v \cdot \nabla_x F = \frac{1}{\mathrm{Kn} } Q(F, F) \end{equation} toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded \begin{equation} \partial_t u + u \cdot \nabla_x u + \nabla_x p = 0, \quad \mathrm{div} u = 0. \end{equation} We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ($\omega \in L^{\mathfrak{p} }$ for any fixed $1 \le \mathfrak{p} < \infty$). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in $L^{\mathfrak{p} }$ as $\mathfrak{p} \rightarrow \infty$ (localized Yudovich class).
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-09-16 13:24:57
Various plasma phenomena will be discussed using a fundamentalfluid model for plasmas, called the Euler-Poisson system. These include plasma sheaths and plasma soliton. First we will briefly introduce recent results on the stability of plasma sheath solutions, and the quasi-neutral limit of the Euler-Poisson system in the presence of plasma sheaths. Another example of ourinterest is plasma solitary waves, for which we discuss existence, stability, and the time-asymptotic behavior. To study the nonlinear stability of solitary waves, the global existence of smooth solutions must be established, which is completelyopen. As a negative answer for global existence, we look into the finite-time blow-up results for the Euler-Poisson system, and discuss the related open questions.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-08-16 09:03:48
In this talk, we propose the Landau-Lifshitz type system augmented with Chern-Simons gauge terms, which can be considered as the geometric analog of so-called the Chern-Simons-Schrodinger equations. We first derive its self-dual equations through the energy minimization so that we can provide $N$-equivariant solitons. We next deliver basic ideas of constructing $N$-equivariant solitary waves for non-self-dual cases and investigating their qualitative properties.
Host: 확률 해석 및 응용 연구센터     Contact: 확률 해석 및 응용 연구센터 (042-350-8111/8117)     미정     2022-08-16 09:01:57