Special Session 45: Partial differential equations from fluids and waves

The qualitative behavior for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

Zhuang Han Harbin Engineering University Peoples Rep of China Co-Author(s):    Runzhang Xu, Yanbing Yang

Abstract: The Cauchy problem for the nonlinear one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity is concerned in this paper. This model describes the propagation of long waves on the surface of water within small amplitude. The main motivation of this paper is to reveal how logarithmic nonlinearity $u \ln |u|^k$ along with the higher-order dispersive term $u_{ x x x x x x }$ affects the qualitative properties of the solution. Some of the efforts on results of global existence and exponential growth of the solution are shown. The main tools to obtain these results include the logarithmic Sobolev inequality, Galerkin method and the concave method. The initial energy is divided into different cases by the depth of the potential well, and corresponding results for subcritical and critical energy levels are both given.

Characteristics of wave propagation in Pre-stressed Viscoelastic Timoshenko Nanobeams with Surface Stress and Magnetic Field Influences

Sunita Kumawat BITS-Pilani, Hyderabad India Co-Author(s):    Sunita Kumawat, Kalyan Boyina, Sumit Kumar Vishwakarma, Raghu Piska

Abstract: The current investigation explores the behavior of pre-stressed viscoelastic Timoshenko nanobeams under the influence of surface effects and a longitudinal magnetic field. Utilizing a modified version of non-local strain gradient theory through the Kelvin-Voigt viscoelastic model, a closed-form dispersion relation using a suitable analytical approach has been derived. To account for surface stresses, Gurtin-Murdouch surface elasticity theory has been employed. Additionally, the study delves into the impact of a longitudinal magnetic field on a single-walled carbon nanotube, considering Lorentz magnetic forces. The validity of the findings is established by deriving results in the absence of surface effects and magnetic fields, aligning well with existing literature. The investigation indicates that pre-stress has marginal effects on flexural and shear waves, while surface effects, magnetic fields, non-locality, characteristic length, and nanotube diameter significantly influence the phase velocity. Additionally, the Threshold velocity and blocking diameter are discussed for the model.

Flexibility results for the Monge-Ampere system

Marta Lewicka University of Pittsburgh USA Co-Author(s):

Abstract: We study flexibility of weak solutions to the Monge-Ampere system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Ampere equation in d=2 dimensions, naturally arising from the prescribed curvature problem and closely related to the classical problem of isometric immersions. Our main results achieve density in the set of subsolutions, of the Holder $\mathcal{C}^{1,\alpha}$ solutions to the Von Karman system which is the weak formulation of (MA). We will present a panorama of recent results in this context, exhibiting regularity dependence on the dimension and codimension of the problem.

Transverse instability of line periodic waves to the KP-I equation

Wei Lian Lund university Sweden Co-Author(s):    Erik Wahlen

Abstract: This is a joint work with Prof. Erik Wahlen (Lund University, Lund, Sweden). The passage from linear instability to nonlinear instability has been shown for 1D solitary waves under 2D perturbations. Although transverse instability of periodic waves to the KdV equation under the KP-I flow has been expected to be true from spectral instability for a long time, it has not been clear how to adapt the general instability theory for solitary waves to periodic waves until now. In this talk, we present how such an adaptation works with the aid of exponential trichotomies and multivariable Puiseux series.

Stability on 3D anisotropic incompressible MHD system near the background magnetic field

Hongxia Lin Chendu University of Technology Peoples Rep of China Co-Author(s):    Jiahong Wu, Yi Zhu

Abstract: The small data global well-posedness of the 3D incompressible Navier-Stokes equations with only one-directional dissipation in the whole space remains an outstanding open problem. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, we investigate the global well-posedness, the stability and large-time behavior of a special 3D magnetohydrodynamic (MHD) system with only one-directional velocity dissipation and horizontal magnetic diffusion near a background magnetic field in the whole space. Firstly, by discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several new techniques to unearth the hidden structure in the nonlinearity, we overcome the derivative loss difficulties and solve the desired global well-posedness and stability problem. Furthermore, by initiating new strategies and developing innovative tools for stability and large-time behavior problems on anisotropic models, we improve the stability to the weaker Sobolev setting . Meanwhile, explicit decay rates are also obtained.

Nonexistence result for diffusion problem with fractional (p,q)-Laplacian at subcritical initial energy level

Qiang Lin Changsha University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: By concavity method and potential well theory, we prove the finite time blowup of the solution for a class of parabolic equations with fractional $(p,q)$-Laplacian at the subcritical initial energy level, which complements a previous work related to the global existence and asymptotic behavior of solutions for the same type of equation.

Global well-posedness of 3D inhomogenous Navier-Stokes system with variable viscosity

Dongjuan Niu Capital Normal University Peoples Rep of China Co-Author(s):    Wang,Lu

Abstract: In this talk, I will present the global well-posedness of 3D inhomogenous Navier-Stokes system with variable viscosity only under the smallness assumptions of initial velocity fields in critical spaces. Compared with the previous results, we remove the small conditions on the initial density by virtue of the velocity decomposition method. In addition, the decay rate of the velocity fields is necessary.

A Cell-Centered Implicit Finite Difference Scheme to Study Wave Propagation in Acoustic Media

Sumit K Vishwakarma Birla Institute of Technology and Science, Pilani India Co-Author(s):    Sunita Kumawat, Ajay Malkoti

Abstract: we present a Cell-Centered Implicit Finite Difference (CCIFD) operator-based numerical scheme for the propagation of acoustic waves that is very effective, accurate, and small in size. This scheme requires fewer estimation points than the traditional central difference derivative operator. Any numerical simulation is significantly impacted by the precision of a numerical derivative. Long stencils can deliver excellent accuracy while also minimising numerical anisotropy error. However, a long stencil requires a lot of computational resources, and as these derivatives get bigger, they could start to look physically unrealistic due to contributions from nodes located extremely far, wherein the derivative is local in nature. Furthermore, using such lengthy stencils at boundary nodes may result in errors. The present article investigates a cell-centered fourth order finite difference scheme to model acoustic wave propagation which utilises a lesser number of nodes in comparison to the traditional Central Difference (CD) operator. However, in general the implicit derivative operator has high computational cost and therefore despite its significant advantages it is generally avoided to be implemented in applications. This serves as a motivation for the present paper to explore a technique called CCIFD that significantly decreases the computational expense by nearly fifty percent. Additionally, spectral characterization of the CCIFD derivative operator has been analysed and discussed. Finally, the wave propagation has been numerically simulated in 2-dimensional homogeneous and Marmousi model using CCIFD scheme to validate the applicability and stability of the scheme.

Well-posedness for p(x)-Laplacian parabolic equations with multiple regime on an annulus

Yitian Wang Harbin Engineering University Peoples Rep of China Co-Author(s):    Runzhang Xu, Chao Yang

Abstract: This study investigates the well-posedness of solutions to the initial boundary value problem for parabolic equations with variable exponents of multiple regime (subcritical, critical, and supercritical) on an annulus. The presence of critical and supercritical regimes disrupts classical Sobolev embeddings, leading to the lack of compactness. To address these issues, we use the Strauss inequality to restore compact Sobolev embeddings for radially symmetric functions. By employing the subdifferential technique with symmetry constraints, we establish local existence of solutions for any radially symmetric initial data and demonstrate uniqueness. We pioneer the application of the potential well theory to classify initial data based on three energy levels: subcritical, critical, and supercritical. For subcritical and critical levels, we analyze cases with non-positive and positive initial energy, obtaining results on finite-time blowup and identifying threshold conditions for global existence versus blowup. Finally, we extend these results to a broader class of locally symmetric domains containing an annulus.