Special Session 12: Hyperbolic Partial Differential Equations and Applications

Unconditional flocking for weak solutions to self-organized systems of Euler-type

Debora Amadori University of L`Aquila Italy Co-Author(s):    Cleopatra Christoforou (University of Cyprus)

Abstract: In this talk, we present some results on the time-asymptotic flocking of weak solutions to a hydrodynamic model of flocking-type with all-to-all interaction kernel, in one-space dimension. An appropriate notion of entropy weak solutions with bounded support is given to capture the behavior of solutions to the Cauchy problem with any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. We will discuss the long time behavior of these solutions, which are shown to experience flocking for large time: their support is uniformly bounded in time, and the velocity converges to the mean value. The rate of convergence is exponential. The proof is based on the decay of positive waves and on cancellation properties between positive and negative waves.

Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$

Lv Cai Shanghai University Peoples Rep of China Co-Author(s):    Ning-An Lai, Wen-Ze Su

Abstract: In this talk, I will report some results on the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in $\R^2$. We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. New vector fields are introduced and better decays for the linear error terms are obtained. This talk is based on a recent joint work with Ning-An Lai and Wen-Ze Su.

Local existence and uniqueness of the strong solution to the heat and moisture transport system in fibrous porous media

Runmei Du Changchun University of Technology Peoples Rep of China Co-Author(s):    Ronghua Pan

Abstract: In this paper, we consider a model coming from textile industries, which can be used to describe the heat and moisture transfer processes in a clothing assembly. The model is a system consisting of two strongly coupled nonlinear parabolic equations in three dimensions, with Robin type boundary conditions. We establish the local existence and uniqueness of the strong solution to the model.

Spectral stability of weak dispersive shocks in quantum hydrodynamics with nonlinear viscosity

Raffaele Folino Universidad Nacional Autonoma de Mexico Mexico Co-Author(s):

Abstract: In this talk, I consider a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics. In particular, the dispersive term is due to quantum effects described through the Bohm potential, while the viscosity term is nonlinear. The main goal is to prove that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable. The proof is based on spectral energy estimates, for which the monotonicity of the profiles in the small-amplitude regime plays a crucial role. This is a joint work with Ramon Plaza (UNAM) and Delyan Zhelyazov (University of Surrey).

Large Time Behaviors of Solutions to the Euler / Euler-Poisson Equations with Time-dependent Damping

Haitong Li Changchun University of Technology Peoples Rep of China Co-Author(s):    Shaohua Chen, Jingyu Li, Ming Mei and Kaijun Zhang

Abstract: In this talk, we consider the Cauchy problem for the 1D Euler / Euler-Poisson equations with time-dependent damping whose coefficient is $-\frac{\mu}{(1+t)^\lambda}$. Firstly, we consider the large time behaviors of solutions to the Euler / Euler-Poisson equations for $\mu=1, -11$ and $\mu>0$, or $\lambda=1$ but $0

Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations

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

Abstract: This talk concerns the 3D forced stochastic Navier-Stokes equation driven by additive noise. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. This is a joint work with Elia Bru$\`e$, Rui Jin, and Deng Zhang.

Entropy bounded solution for liquid crystal hydrodynamics

Tariq Mahmood Northwestern Polytechnical University, Xian Peoples Rep of China Co-Author(s):

Abstract: The research on liquid crystal hydrodynamics has been a hot topic for the last few decades, due to its large-scale applications in real-world problems. In this work, we survey two theories, Ericksen-Leslie and Landau De Gennes, and their progress in the analysis of partial differential equations. Our main focus will be on uniaxial theory E-L and their mathematical results, namely the global well-posedness of strong solutions to the liquid crystal equations, and entropy bounds as long as the initial density decreases gradually at the far field. We will also comment on the interesting mathematical problems from the Q-tensor representation (Landau De Gennes theory), which is more complex than Ericksen-Leslie`s theory.

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.

Global large smooth solutions and relaxation limit of isothermal Euler equations

Richard Yue-Jun RYJ Peng University of Clermont Auvergne France Co-Author(s):

Abstract: In this talk, I show that the Cauchy problem for isothermal Euler equations with relaxation admits a unique global smooth solution when either the relaxation time or the initial datum is sufficiently small. The large smooth solution is then obtained when the relaxation time is sufficiently small. Moreover, I establish error estimates for the convergence of the large density of the Euler equations toward the solution of the heat equation as the relaxation time tends to zero. Besides classical energy estimates, a uniform estimate for a quantity related to Darcy`s law is important in the proof of the above results.

Vanishing Shear Viscosity and Boundary Layer for the Navier-Stokes Equations with Cylindrical Symmetry and Planar MHD system

Xulong Qin Sun Yat-sen University Peoples Rep of China Co-Author(s):

Abstract: We consider an initial boundary problem for the Navier-Stokes Equations with Cylindrical Symmetry and the planar MHD system . The limit of the vanishing shear viscosity is justified. In addition, the $L^2$ convergence rate is obtained together with the estimation on the thickness of the boundary layer.

Global Existence and Convergence of Large Strong Solutions to the 3D Full Compressible Navier Stokes Equations

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

Abstract: In this talk we consider the Cauchy problem of global in time existence of large strong solutions to the Navier Stokes equations for compressible viscous and heat conducting fluids. A class of density dependent viscosity is considered. By introducing the modified effective viscous flux and using the bootstrap argument, we establish the global existence of large strong solutions when the initial density is linearly equivalent to a large constant state. It should be mentioned that this result is obtained without any restrictions on the size of initial velocity and initial temperature. In addition, we establish the convergence of the solutions to its associated equilibrium with an explicit decay rate.

Stability of stationary solutions for viscoelastic fluids in half-space

Yoshihiro Ueda Kobe University Japan Co-Author(s):    Yusuke Ishigaki

Abstract: In this talk, we discuss the stability of the compressible fluid with viscoelasticity. We consider the outflow problem in a one-dimensional half-space and show the existence of a stationary solution and its stability. There exists a lot of known results for compressible fluids. In particular, the existence and stability of stationary solutions to the outflow problem were discussed in Nakamura-Nishibata-Yuge (2007) and Nakamura-Ueda-Kawashima (2010), where the Mach number was used as a criterion. Similar results are obtained for viscoelastic fluids, however, the main feature is that the criterion is constructed by the modified Mach number, which takes into account the effect of viscoelasticity. This result is based on joint research with Yusuke Ishigaki of Osaka University.