Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, December 1, 2015 - 15:05 , Location: Skiles 006 , Young-Pil Choi , Imperial College London , Organizer: Wilfrid Gangbo
The interactions between particles and fluid have received a bulk of attention due to a number of their applications in the field of, for example, biotechnology, medicine, and in the study of sedimentation phenomenon, compressibility of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this talk, we present coupled hydrodynamic equations which can formally be derived from Vlasov-Boltzmann/Navier-Stokes equations. More precisely, our proposed equations consist of the compressible pressureless Euler equations and the isentropic compressible Navier-Stokes equations. For the coupled system, we establish the global existence of classical solutions when the domain is periodic, and its large-time behavior which shows the exponential alignment between two fluid velocities. We also remark on blow-up of classical solutions in the whole space.
Series: PDE Seminar
Tuesday, November 17, 2015 - 15:05 , Location: Skiles 006 , Geng Chen , School of Mathematics, Georgia Tech , Organizer: Wilfrid Gangbo
In this talk, we will discuss a sequence of recent progresses on the global well-posedness of energy conservative Holder continuous weak solutions for a class of nonlinear variational wave equations and the Camassa-Holm equation, etc. A typical feature of solutions in these equations is the formation of cusp singularity and peaked soliton waves (peakons), even when initial data are smooth. The lack of Lipschitz continuity of solutions gives the major difficulty in studying the well-posedness and behaviors of solutions. Several collaboration works with Alberto Bressan will be discussed, including the uniqueness by characteristic method, Lipschitz continuous dependence on a Finsler type optimal transport metric and a generic regularity result using Thom's transversality theorem.
Series: PDE Seminar
Tuesday, November 10, 2015 - 15:05 , Location: Skiles 006 , Hyung Ju Hwang , POSTECH, Korea , Organizer: Wilfrid Gangbo
In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang,J. Jung, and J. Velazquez.
Series: PDE Seminar
Thursday, October 22, 2015 - 15:05 , Location: Skiles 005 , Hermano Frid , Institute de Matematica Pura e Aplicada (IMPA) , Organizer: Ronghua Pan
We consider a Benney-type system modeling short wave-long wave interactions in compressible viscous fluids under the influence of a magnetic field. Accordingly, this large system now consists of the compressible MHD equations coupled with a nonlinear Schodinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. An important point here is that, instead of the simpler case having zero as the equilibrium state for the magnetic field, we consider an arbitrary non-zero equilibrium state B for the magnetic field. This is motivated by applications, e.g., Earth's magnetic field, and the lack of invariance of the MHD system with respect to either translations or rotations of the magnetic field. The usual time decay investigation through spectral analysis in this non-zero equilibrium case meets serious difficulties, for the eigenvalues in the frequency space are no longer spherically symmetric. Instead, we employ a recently developed technique of energy estimates involving evolution in negative Besov spaces, and combine it with the particular interplay here between Eulerian and Lagrangian coordinates. This is a joint work with Junxiong Jia and Ronghua Pan.
Series: PDE Seminar
Tuesday, October 20, 2015 - 15:05 , Location: Skiles 006 , Nestor Guillen , University of Massachusetts at Amherst , Organizer: Wilfrid Gangbo
The study of reflector surfaces in geometric optics necessitates the analysis of nonlinear equations of Monge-Ampere type. For many important examples (including the near field reflector problem), the equation no longer falls within the scope of optimal transport, but within the class of "Generated Jacobian equations" (GJEs). This class of equations was recently introduced by Trudinger, motivated by problems in geometric optics, however they appear in many others areas (e.g. variations of the Minkowski problem in convex geometry). Under natural assumptions, we prove Holder regularity for the gradient of weak solutions. The results are new in particular for the near-field point source reflector problem, but are applicable for a broad class of GJEs: those satisfying an analogue of the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. Joint work with Jun Kitagawa.
Series: PDE Seminar
Tuesday, October 6, 2015 - 15:05 , Location: Skiles 006 , Ryan Hynd , University of Pennsylvania , Organizer: Wilfrid Gangbo
The smallest eigenvalue of a symmetric matrix A can be expressed through Rayleigh's formula. Moreover, if the smallest eigenvalue is simple, it can be approximated by using the inverse iteration method or by studying the large time behavior of solutions of the ODE x'(t)=-Ax(t). We discuss surprising analogs of these facts for a nonlinear PDE eigenvalue problem involving the p-Laplacian.
Series: PDE Seminar
Tuesday, September 8, 2015 - 15:05 , Location: Skiles 006 , Chongchun Zeng , School of Mathematics, Georgia Tech , Organizer: Wilfrid Gangbo
Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that R(L) is closed, L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded and with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^{tJL}. More specifically, we obtain some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^{tJL}. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases (particularly ground states), we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This work will be discussed in two talks. In the first talk, we will motivate the problem by several Hamiltonian PDEs, describe the main results, and demonstrate how they are applied. In the second talk, some ideas of the proof will be given.
Series: PDE Seminar
Tuesday, September 1, 2015 - 15:05 , Location: Skiles 006 , Zhiwu Lin , School of Mathematics, Georgia Tech , Organizer: Wilfrid Gangbo
Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that R(L) is closed, L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded and with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^{tJL}. More specifically, we obtain some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^{tJL}. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases (particularly ground states), we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This work will be discussed in two talks. In the first talk, we will motivate the problem by several Hamiltonian PDEs, describe the main results, and demonstrate how they are applied. In the second talk, some ideas of the proof will be given.
Series: PDE Seminar
Tuesday, August 25, 2015 - 15:05 , Location: Skiles 006 , Shihui Zhu , Department of Mathematics, Sichuan Normal University , Organizer: Wilfrid Gangbo
In this talk, we consider the dynamical properties of solutions to the fractional nonlinear Schrodinger equation (FNLS, for short) arising from pseudorelativistic Boson stars. First, by establishing the profile decomposition of bounded sequences in H^s, we find the best constant of a Gagliardo-Nirenberg type inequality. Then, we obtain the stability and instability of standing waves for (FNLS) by the profile decomposition. Finally, we investigate the dynamical properties of blow-up solutions for (FNLS), including sharp threshold mass, concentration and limiting profile. (Joint joint with Jian Zhang)
Series: PDE Seminar
Tuesday, August 18, 2015 - 15:05 , Location: Skiles 006 , Yi Wang , AMSS, Chinese Academy of Sciences , Organizer: Wilfrid Gangbo
We investigate the nonlinear stability of elementary wave patterns (such as shock, rarefaction wave and contact discontinuity, etc) for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the elementary wave patterns to the system. Then, the time-asymptotic stability of the planar rarefaction wave, viscous shock waves and viscous contact wave (viscous version of contact discontinuity) are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. These results imply that these basic wave patterns are still stable in the transportation of charged particles under the binary collision, mutual interaction, and the effect of the electrostatic potential force. The talk is based on the joint works with Hailiang Li (CNU, China), Tong Yang (CityU, Hong Kong) and Mingying Zhong (GXU, China).

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