Abstract : In 1979, Gidas, Ni, and Nirenberg popularized the moving plane method, which is useful for demonstrating the radial symmetry and monotonicity of positive smooth solutions to specific elliptic equations in radially symmetric domains. We will briefly review some works following their 1979 contributions. Afterward, we will explain how the moving plane method implies radial symmetry and present a simplified new proof for a specific case introduced by Chen and Li in 2017.
Reference :
(직접 참고한 강의록) Dana Berman ' The Method of Moving Planes and Applications to Elliptic Equations '(2019)
(언급할 연구결과)
B. Gidas , Wei-Ming Ni , and L. Nirenberg 'Symmetry and Related Properties via the Maximum Principle'(1979)
B. Gidas , Wei-Ming Ni , and L. Nirenberg 'Symmetry of Positive Solutions of Nonlinear Elliptic Equations in R^{n}'(1981)
Wenxing Chen, Conming Li 'Classification of solutions of some nonlinear elliptic equations'(1991)
Abstract: We consider the Euler-Poisson system, which describes the ion dynamics in electrostatic plasmas. In plasma physics, the pressureless model is often employed to simplify analysis. However, the behavior of solutions to the pressureless model generally differs from that of the isothermal model, both qualitatively and quantitatively - for instance, in the case of blow-up solutions.
In previous work, we investigated a class of initial data leads to finite-time C^1 blow-up solutions. In order to understand more precise blow-up profiles, we construct blow-up solutions converging to the stable self-similar blow-up profile of the Burgers equation. For the isothermal model, the density and velocity exhibit C^{1/3} regularity at the blow-up time. For the pressureless model, we provide the exact blow-up profile of the density function, showing that the density is not a Dirac measure at the moment of blow-up.
We also consider the peaked traveling solitary waves, which are not differentiable at a point. Our findings show that the singularities of these peaked solitary waves have nothing to do with the Burgers blow-up singularity. We study numerical solutions to the Euler-Poisson system to provide evidence of whether there are solutions whose blow-up nature is not shock-like.
This talk is based on collaborative work with Junho Choi (KAIST), Yunjoo Kim, Bongsuk Kwon, Sang-Hyuck Moon, and Kwan Woo (UNIST).
**Abstract**:
In 1960, Il’in and Oleinik analyzed the asymptotic behavior of solutions to the Riemann problem for the hyperbolic part of a one-dimensional scalar viscous conservation law with prescribed far-field states. In the mid-1980s, Matsumura-Nishihara and Goodman independently discovered that the \(L^2\)-energy method could be applied to study the asymptotic stability of viscous shock waves. Later, Kang and Vasseur further developed this method and successfully applied it to specific systems and scalar equations. In this seminar, we will briefly review the works of Matsumura-Nishihara and Kang-Vasseur, focusing on how the \(L^2\)-energy method can be applied to scalar equations and providing a brief proof.
**References**:
1. Matsumura, A., & Nishihara, K. (1994). Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. *Communications in Mathematical Physics*, 165, 83-96.
2. Kang, M. J., & Vasseur, A. F. (2017, January). L2-contraction for shock waves of scalar viscous conservation laws. In *Annales de l'Institut Henri Poincaré C, Analyse non linéaire* (Vol. 34, No. 1, pp. 139-156).
Abstract:
Regularity theory has been a long-standing research area in partial differential equations(PDEs).
A classic result in many textbooks is the interior regularity, or Schauder estimates, for elliptic PDEs with continuously differentiable or Hölder continuous coefficients.
As a relatively recent result, global $W^{1,\infty}$ and piecewise $C^{1,\alpha}$ estimates have been achieved for solutions to divergence form elliptic equations with piecewise H\"{o}lder continuous coefficients.
This talk will present an overview of that result and its application.
Reference:
Li, Yan Yan, and Michael Vogelius. "Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients." Archive for rational mechanics and analysis 153 (2000): 91-151.
Li, Yanyan, and Louis Nirenberg. "Estimates for elliptic systems from composite material." Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 56.7 (2003): 892-925.
Ammari, Habib, et al. "Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case." Journal de mathématiques pures et appliquées 94.3 (2010): 322-339.
초록: In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.
We examine the dynamics of short-range interacting Bose gases with varying diluteness and interaction strength. Using a combination of mean-field and semiclassical methods, we show that, for large numbers of particles, the system’s local mass, momentum, and energy densities can be approximated by solutions to the compressible Euler system (with pressure P = gρ2 ) up to a blow-up time. In the hard-core limit, two key results are presented: the internal energy is derived solely from the many-body kinetic energy, and the coupling constant g = 4πc0 where c0 the electrostatic capacity of the interaction potential. The talk is based on our recent work arXiv:2409.14812v1. This is joint work with Shunlin Shen and Zhifei Zhang. The talk will be delivered in English and is meant for the general audience.
In the N-body problem, choreographies are periodic solutions where N equal masses follow each other along a closed curve. Each mass takes periodically the position of the next after a fixed interval of time. In 1993, Moore discovered numerically a choreography for N = 3 in the shape of an eight. The proof of its existence is established in 2000 by Chenciner and Montgomery. In the same year, Marchal published his work on the most symmetric family of spatial periodic orbits, bifurcating from the Lagrange triangle by continuation with respect to the period. This continuation class is referred to as the P12 family. Noting that the figure eight possesses the same twelve symmetries as the P12 family, the author claimed that it ought to belong to P12. This is known as Marchal’s conjecture. In this talk, we present a constructive proof of Marchal’s conjecture. We formulate a one parameter family of functional equations, whose zeros correspond to periodic solutions satisfying the symmetries of P12; the frequency of a rotating frame is used as the continuation parameter. The goal is then to prove the uniform contraction of a mapping, in a neighbourhood of an approximation of the family of choreographies starting at the Lagrange triangle and ending at the figure eight. The contraction is set in the Banach space of rapidly decaying Fourier-Chebyshev series coefficients. While the Fourier basis is employed to model the temporal periodicity of the solutions, the Chebyshev basis captures their parameter dependence. In this framework, we obtain a high-order approximation of the family as a finite number of Fourier polynomials, where each coefficient is itself given by a finite number of Chebyshev polynomials. The contraction argument hinges on the local isolation of each individual choreography in the family. However, symmetry breaking bifurcations occur at the Lagrange triangle and the figure eight. At the figure eight, there is a translation invariance in the normal direction to the eight. We explore how the conservation of the linear momentum in this direction can be leveraged to impose a zero average value in time for the choreographies. Lastly, at the Lagrange triangle, its (planar) homothetic family meets the (off-plane) P12 family. We discuss how a blow-up (as in “zoom-in”) method provides an auxiliary problem which only retains the desired P12 family.
초록: In this talk, we consider the self-dual O(3) Maxwell–Chern–Simons-Higgs equation, a semilinear elliptic system, defined on a flat two torus. We discuss about pointwise convergence behavior, which represents the Chern-Simons limit behavior of our system. Building upon this observation, we study the existence, stability, and asymptomatic behavior of solutions.
Abstract: The Hardy's inequality is a classical result in analysis, providing a relationship between the weighted norms of functions and their gradient norm, with several applications in Sobolev spaces, partial differential equations, and mathematical physics. In recent decades, interest has grown in fractional versions of this inequality, where the classical Laplacian is replaced by a non-local operator, such as the fractional Laplacian.
In this talk, we investigate the difficulty in applying the methods used to derive the classical Hardy's inequality to the setting of non-local operators, and introduce a new method to address these challenges, namely general Hardy's inequality. Furthermore, we will examine the relationship with classical results by taking limits of the fractional differential order in the sharp fractional Hardy's inequality.
초록 : The KP equation is a generalization to two spatial dimensions of the one-dimensional KdV equation. We study the eigenvalue problems for the KP-I and KP-II around one-dimensional KdV solitary waves. We show that there is an exponentially growing unstable mode for the KP-I. For KP-II, we find that there are no exponentially unstable modes.
Reference : On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation (J. Alexander, R. Pego, R. Sachs, 1997, Physics Letters A)
Abstract: In this talk, we will discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. This is a joint work with Luigi De Rosa (University of Basel).
Abstract.