Research On The Dynamic Behavior Of Nonlinear Dynamical Systems In Fluid Mechanics

The thesis investigates several types of nonlinear dynamical systems in fluids separately.These dynamical systems are:nonlinear water wave system describing shallow water wave propagation,hydrodynamic system describing active liquid crystal fluids,wave system describing wave phenomena,general fourth-order wave system used to study phenomena such as resonance interaction mechanisms between waves,and pseudoparabolic system describing unsaturated flow processes in porous media.Based on the existing work,the dynamics of these systems under specific initial conditions are studied.And asymptotic stability,global wellposedness,exponential decay and blowup results are obtained respectively.The second chapter of this thesis focuses on a type of quasilinear dispersion equation(Novikov equation)describing a nonlinear water wave system,and establishes the asymptotic stability of the peaked solitary wave solution of the system.This water wave system is used to model the propagation of relatively large amplitude shallow water waves,allowing water waves with stronger nonlinearity than the classic KdV equation.This thesis considers weak solutions with non-negative momentum density,and proposes a new way to describe the localization of weak solutions.The localized structure of solutions helps build the rigidity of the localized solutions.Based on the given orbital stability result and the rigidity of the peaked solitary wave,we further use the modulation argument to prove that the peaked solitary wave solution is asymptotically stable in the energy space.The third chapter of this thesis focuses on the incompressible flow of a class of active liquid crystal fluid systems with inhomogeneous density.This system is used to describe the nematic liquid crystals that have many applications in synthetic chemistry and materials science,etc.We adopt Landau-de Gennes Q-tensor description,that is,the nematic state is described by the order parameters of the Q-tensor,the approximate solution is constructed based on the Faedo-Galerkin method,and then the two levels of approximations are used to establish the global weak solution of the system in the bounded region.In order to use the energy method,considering the nonconstant positive energy density of Landau-de Gennes,we redefine the constant positive energy to obtain the critical estimate of the Q-tensor.Due to the strong nonlinearity of the active term,the weak convergence in the two levels of approximations depends on the cancellation established by the structure of the system and the symmetry of the Q-tensor.With the above results,we finally derive compactness estimates,which prove the existence of the global weak solutions.The fourth chapter of this thesis focuses on a semilinear wave system with logarithmic nonlinear source terms.Logarithmic nonlinear sources originally appear in inflation cosmology and supersymmetric field theory and later were found in different physical backgrounds,including nuclear physics,optics and geophysics.The thesis aims at using potential well theory to study the global existence and non-existence(finite time blow up)of solutions to the system at different initial energy levels.With the help of the logarithmic Sobolev inequality,we establish a variational framework for the wave system with the logarithmic source term which is under a polynomial structure,then derive the global existence,and infinite time blowup of the solutions at the low initial energy level and critical initial energy level.Due to the failure of the invariant set at the supercritical energy level,we find a similar suitable initial condition to establish an infinite time blowup of the solutions.The fifth chapter of this thesis conducts a comprehensive study on the wave system with fourth-order strong dissipation and generalized source term.In order to supplement the remaining problems of the strain wave system within the potential well theory,we consider various terms(fourth-order term,strong damping term,nonlinear damping term and strain term)in the existing literature and incorporate them into the model equation.Also,the polynomial source term is extended to a more general case.First,the existence of the local solution of the system is established.With the aid of the variational tools of potential well theory,we systematically establish the global existence and non-existence of the solution at several energy levels.The sixth chapter of this thesis focuses on a heat system with a generalized source term and a strong damping term used to describe the diffusion phenomenon in the porous media.We first use the Faedo-Galerkin method and contraction mapping principle to prove the existence and uniqueness of the local-in-time solution.Then the potential well family is introduced through the establishment of the variational structure.The dynamic behaviors of the solutions to the system at different initial energy levels are discussed.For the low initial energy level and critical energy level,it is derived that the global existence and exponential decay of the solution and the finite time blowup and the upper bound of blowup time of solution.We obtain the finite time blowup and the upper bound of blowup for the supercritical energy level.In order to overcome the difficulties caused by the singular term,we introduce an approximate weight function to eliminate the singular term,further use Hardy-Sobolev inequality to get critical estimates,and finally establish the local existence of the solution.