Special Session 13: Hyperbolic Partial Differential Equations and Applications

Blow-up mechanism for 2-D compressible MHD equations with radial symmetry

Lv Cai Shanghai University Peoples Rep of China Co-Author(s):    Liu Jianli, Sheng Wancheng

Abstract: We investigate the gradient blow-up mechanism of classical solution to the initial value problem of radially symmetric MHD equations in two spatial dimensions, where the inhomogeneity caused by polar radius brings new difficulties in the high-dimensional problem. By developing delicate estimates for the Riemann variables with coupling effects introduced by the magnetic field, we first obtain the upper bound for the solution itself under compressive initial conditions. Furthermore, we also get the blow-up of first derivatives of solution by constructing the Riccati equations. This generalizes the result of G. Chen, R. Young and Q. Zhang`s work for 1-D MHD with orthogonal magnetic field [J. Hyperbolic Differ. Equ. 10 (2013), 149-172].

Local-in-time well-posedness for the regular solution to the 2D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity

Yue Cao East China University of Science and Technology Peoples Rep of China Co-Author(s):    Xun Jiang

Abstract: We consider the two-dimensional Cauchy problem of the full compressible Navier-Stokes equations with far-field vacuum in $\mathbb{R}^2$, where the viscosity and heat-conductivity coefficients depend on the absolute temperature $\theta$ in the form of $\theta^\nu$ with $\nu>0$. Due to the appearance of the vacuum, the momentum equation are both degenerate in the time evolution and spatial dissipation, which makes the study on the well-posedness challenging. By establishing some new singular-weighted (negative powers of the density $\rho$) estimates of the solution, we establish the local-in-time well-posedness of the regular solution with far-field vacuum in terms of $\rho$, the velocity $u$ and the entropy $S$. Moreover, the uniform regularity of the entropy $S$ in $\mathbb{R}^2$ is established, i.e., $S-\bar{S}\in C([0,T];H^3(\mathbb{R}^2))$ for some constant $\bar{S}$.

Singularity Formation for Supersonic Inward Wave of Radially Symmetric Euler Equations with Damping

Haitong Li Changchun University of Technology Peoples Rep of China Co-Author(s):    Jiabin Li, Qiang Zou

Abstract: This talk focuses on the singularity formation for smooth solutions to the compressible radially symmetric Euler equations with damping. A smooth solution is called a supersonic inward wave if it satisfies $u<-h<0$. For polytropic gases with $\gamma\geq 3$, the damping term introduces additional negative contributions into the evolution equations, which significantly modifies the structure of the corresponding Riccati type inequalities and accelerates the growth of gradients. By applying the characteristic method and constructing suitable invariant domains which different from the case of expanding wave, we prove that smooth supersonic inward solutions with sufficiently strong initial compression must develop singularity in finite time. Furthermore, an explicit upper bound for the lifespan of solutions is derived, which quantitatively characterizes the blowup mechanism of inward wave under damping. This work develops the studies by G. Chen et al. (arXiv: 2511.15180, 2025).

Non-uniqueness of weak solutions for the hypo-viscous compressible Navier-Stokes equations

Yachun Li Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations under general pressure laws. By using convex integration method we provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Due to the difficulties caused by the relative rigidity of the pressure and by the viscosity, our proof features new constructions of building blocks for both the density and momentum, which are, particularly, designed to respect the compressible structure. We provide first applications of a temporally intermittent but spatially homogeneous building block scheme and a mild density perturbation. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations. This is a joint work with Peng Qu, Zirong Zeng, and Deng Zhang.

Formation of singularities for the relativistic membrane equation with radial symmetry

Jianli Liu Shanghai university Peoples Rep of China Co-Author(s):

Abstract: The relativistic membrane equation can be rewritten as a first order hyperbolic system. Making use of the characteristic decomposition method, a new blow-up theorem is established. It demonstrates the formation of singularities for the relativistic membrane equation. Indeed, the singularity occurs when the hypersurface turns from being timelike to being null. This work is cooperated with Dr. Cai Lv of Shanghai University.

Stability of viscous shock for Burgers equations with double singularities in viscosity and flux

Ming Mei Jiangxi Normal University Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with Burgers equation with singular viscosity and singular flux. We realize that, when the singularity for flux is less than the singularity of viscosity, there exists smooth viscous shock wave, otherwise the shock wave does not exist. The main issue is to show the stability of these shock waves. To overcome the singularities caused by viscosity and flux, we use the weighted energy method, where the selection of weights are technical and play a crucial role in the proof.

Rayleigh-Taylor instability and beyond

Ronghua Pan Georgia Institute of Technology USA Co-Author(s):

Abstract: It is known in physics that steady state of fluids under the influence of uniform gravity is stable if and only if the convection is absent. In the context of incompressible fluids, convection happens when heavier fluids is on top of lighter fluids, known as Rayleigh-Taylor instability. However, in real world, heat transfer plays an important role in convection of fluids, such as the weather changes, and or cooking a meal. In this context, the compressibility of the fluids becomes important. Indeed, using the more realistic model of compressible flow with heat transfer, the behavior of solutions is much closer to the real world and more complicated. We will discuss these topics in this lecture, including some on-going research projects.

Supersonic reacting jet flows from a three-dimensional conical nozzle

Wancheng Sheng Shanghai University Peoples Rep of China Co-Author(s):

Abstract: We study supersonic reacting jet flows from a three dimensional (3D) divergent conical nozzle. The flow is governed by 3D steady Zeldovich-von Neumann-D\oring combustion equations with cylindrical symmetry and that the state of the flow is given at the inlet of the nozzle. When the nozzle is surrounded by a vacuum, a global continuous and piecewise smooth supersonic reacting jet flow expanding into the vacuum from the nozzle is obtained. When the nozzle is surrounded by a static atmosphere with a lower pressure than the pressure of the flow at the exit of the nozzle, we obtained a local continuous and piecewise smooth supersonic reacting jet flow expanding into the atmosphere from the nozzle. Moreover, an explanation for the formation of intercepting shocks in supersonic jets expanding into a lower pressure environment is shown, which is stated in the book Supersonic Flow and Shock Waves and is verified by physical experiments. The flow patterns constructed in the paper may be used as background solutions for more general supersonic jet flow problems. This work is jointed with Prof. G.Lai.

Study of fluid equations in non-smooth domains

Yaguang Wang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    WANG Yaguang

Abstract: In this talk, we shall study the well-posedness of the initial boundary value problem for the two-dimensional compressible ideal magnetohydrodynamic equations with the impermeable and perfectly conducting boundary conditions in domains with corners. Meanwhile, the small viscosity limit of viscous flow in quarter space will also be investigated. It is based on joint works with Wen Guo.

Global existence of large solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent coefficients and vacuum

Shang Zhaoyang Shanghai Lixin University of Accounting and Finance Peoples Rep of China Co-Author(s):    Yachun Li, Peng Lu, Wei Xiang

Abstract: It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this talk, we study the 3D Navier-Stokes equations with temperature-dependent coefficients in the whole space, and establish the first result on the global existence of large strong solution when the initial density and the initial temperature are linearly equivalent to some large constant states. Moreover, the optimal decay rates of the solution to its associated equilibrium are established when the initial data belong to $L^{p_0}(\R^3)$ for some $p_0\in[1,2]$.

Global small solutions to the 2D non-resistive compressible MHD system near an equilibrium

Yi Zhu East China University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: This talk focuses on the compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in $\mathbb{R}^2$. We present a systematic approach to establish the global existence of smooth solutions when the initial data is close to a background magnetic field. To overcome key challenges arising from anisotropy and the critical nature of the system, we fully exploit the underlying dissipative structure and introduce several delicately constructed dissipative quantities. Compared with the previous works, we solve this problem by pure energy estimate without extra help from Sobolev spaces of negative index