Special Session 194: The rigirous mathematical theory on the convergence in the fluid related model

Global well-posedness of one-dimensional infrarelativistic model for a compressibleviscous gas with radiation

Xin Liu Shanghai University of International Business and Economics Peoples Rep of China Co-Author(s):    Yuming Qin, XIaolei Dong

Abstract: This talk investigates the global well-posedness of the one-dimensional infrarelativistic model for a compressible viscous gas with radiation. Under suitable hypotheses on the transport coefficients, we prove the global existence and asymptotic behavior of solutions in H^i (i = 1, 2, 4) for this model.

Enhanced Dissipation and Global Well-Posedness for a Three-Dimensional Flame Propagation Model with Couette Flow

Lijuan Wang Shanghai University of International Business and Economics Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with a three-dimensional gravity-induced flame front model under a Couette flow. By exploiting the enhanced dissipation induced by the Couette flow, we prove global-in-time well-posedness of the Cauchy problem in $\mathbb{R}^3$ and derive decay estimates for the solution and its spatial derivatives in $L^p$ norms for all $p \ge 1$. The analysis is based on a Green`s function approach for the associated variable-coefficient linearized operator.These results show that enhanced dissipation induced by the Couette flow is the key mechanism leading to global existence in the whole space, in the large initial data regime. This talk is based on recent joint work with Professor Yoshiyuki Kagei

Suppression of blow-up in 3-D Keller-Segel system with fractional diffusion via Couette flow in whole space

Yucheng Wang Shanghai University Peoples Rep of China Co-Author(s):    Shijin Deng, Binbin Shi, Weike Wang

Abstract: In this paper, we consider a Keller-Segel model with a fractional diffusion term in $\mathbb{R}^3$ in the background of a Couette flow. We show that when the background Couette flow is large enough, the dissipation enhancement induced could prevent the blow-up of solutions and thus prove the global existence and also obtain time decay rates of the solution in $L^p$ norm. The main tool of the proof is a corresponding Green`s function and the key estimate is its $L^1$ estimate without singularities at $t=0$. To fulfill such an estimate, we meet great troubles caused by the fractional heat kernel together with the Couette flow in the model considered here and overcome the troubles by introducing a space-frequency mixed decomposition. The main purpose of this paper is to develop a new microlocal analysis technique for researching the Green's functions with variable coefficients and strong singularities, such as the Green's function with shear flows. Obviously, this technique is crucial for capturing the dissipation enhancement mechanism of shear flows in the whole space.

Decoupled rotational Camassa-Holm approximation from the Green-Naghdi system with Coriolis effect

Xiongfeng Yang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Yue Liu

Abstract: The present paper demonstrates the rigorously convergence from the Green-Naghdi (R-GN) system incorporating Coriolis effects to two counter-propagating wave packets governed by distinct rotational Camassa-Holm (R-CH) equations within the Camassa-Holm framework. The crucial observation is the construction of appropriate approximation functions to precisely characterize the nonlinear interaction dynamics between these bidirectional wave components. This mathematical framework extends the Camassa-Holm paradigm to rotational flows through careful modulation of wave dispersion and Coriolis coupling effects.