Our seminar takes place every Thursday at 9am in Room 114, Math Building.

In general, each presentation lasts 45 minutes, with a discussion session afterwards.

Time plan of talks:

Sep 18 -- Mengyu Cheng (程梦雨, Associate Prof. @DUT)

Title: Random attractors for SDEs

Abstract: The attractor is an important concept in dynamical systems, which can be used to characterize the long time asymptotic behavior of systems. In this talk, we will discuss random attractors for McKean-Vlasov stochastic differential equations (in short, MVSDEs). A difficulty arises from the distribution dependence in MVSDEs, which breaks the flow property of solutions. To address this, we first consider the random attractor on the product space $H \times \mathcal{P}(H)$. Then we establish the existence of random attractors on $H$. Here, $H$ is a separable Hilbert space and $\mathcal{P}(H)$ denotes the space of probability measures on $H$.

Sep 25 -- Jie Fan (樊洁, PostDoc @ AMSS, CAS)

Title: Global Well-Posedness and Singularity Analysis for Solutions of Compressible Fluids

Abstract: This talk primarily presents singularity criteria for solutions to the compressible Navier-Stokes equations. It further extends a previously proven conjecture of Nash. Another part establishes the global existence of solutions for the MHD equations where the viscosity coefficients depend on the density. In this case, the initial density can be arbitrarily large.

Oct 9 -- Lili Wang (王莉莉, Ph. D. student @ DUT)

Title: Blow-up suppression for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow

Abstract: As is well-known, the solution of the Patlak-Keller-Segel system in 3D will blow up in finite time regardless of any initial cell mass. 

In this talk, we are interested in the suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the stabilizing effect of the moving fluid. We prove that if the Couette flow is sufficiently strong, then the solutions for the system are global in time. This is a joint work with Shikun Cui, Wendong Wang and Juncheng Wei.

Oct 16 -- Two talks

9:00--10:00 Shikun Cui (崔世坤, Ph. D. student @ DUT)

Title: Stability and instability of Standing Periodic Waves in the Massive Thirring Model

Abstract: In this talk, we study the spectral stability of the standing periodic waves in the massive Thirring model. The Massive Thirring Model is complete integrable, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. We show analytically that each family of standing periodic waves is distinguished by the location of eight eigenvalues which coincide with the end points of the spectral bands of the Lax spectrum. The standing periodic waves are proven to be spectrally stable if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane. By computing the Lax spectrum numerically, we show that this stability criterion is satisfied for some standing periodic waves. This is joint work with Prof. Dmitry Pelinovsky.

10:00--11:00 Yang Cao (曹杨, Prof. @ DUT)

Title: Threshold dynamics of traveling waves for monostable pseudo-parabolic equation

Abstract: This report is about the traveling wave solutions to the pseudo-parabolic equation, a kind of non-classical diffusion equation characterized by the mixed third-order derivative term. We demonstrate that the ratio of the mixed third-order derivative coefficient to the diffusion coefficient $\frac{\tau}{D}$ can serve as a bifurcation parameter. In detail, when $\frac{\tau}{D}\leq 1$, the equation possesses monotone traveling waves; when $\frac{\tau}{D}>1$, traveling waves are not monotonic and oscillate around the steady state $u=1$. The precise form of the minimal wave speed $c^*(\tau,D)$ is also derived, exhibiting a monotonic increase with respect to $\tau$ and converging to $2\sqrt{D}$ as $\tau$ approaches $0$. Numerical simulations confirm and support our theoretical results. They further show that the larger the value of $\tau$ is, the more non-monotonic the traveling waves become. Our findings regarding oscillating traveling waves predict saturation overshoot---a behavior that contradicts classical diffusion-like behavior yet is widely observed in unsaturated porous media. Mathematically, the threshold value of $\frac{\tau}{D}$ reveals the essential role of the dynamic capillary effect in the fundamental overshoot mechanism.

Oct 23 -- Junhyeok Byeon (Associate Prof. @DUT)

Title: TBA