On a class of quasilinear Schrodinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sutficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.     

Diffusion-Induced Blowup in a Nonlinear Parabolic System

A two-component semilinear parabolic system on a bounded domain with Neumann boundary conditions is studied. It is shown that for a certain kind of nonlinearity, the blowup of solutions may occur when the diffusion coefficients are not equal, though the corresponding ODE possesses a globally stable equilibrium.     

Global solution for coupled nonlinear Klein-Gordon system

The globed solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.     

Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift–Diffusion Type

We consider a Cauchy problem for a parabolic-elliptic system of drift–diffusion type. The problem is formally of the form

$ U_t = \nabla \cdot (\nabla U-U \nabla (-\Delta)^{-1}U). $

This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I, in the sense that its blowup rate is the same as the corresponding ordinary differential equation y _t  = y ² (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar, provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three.

     

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method.Moreover,it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.     

High-symmetry Kida flow: Time series analysis and resummation

A high-symmetry Kida flow is studied as a candidate for a finite time blowup of incompressible Euler equations. Explicit formulas for the solutions of the Euler equations, in the class of formal power series, are derived after transforming the momentum equation into a homogeneous quadratic differential equation on a nonassociative algebra. Using these formulas, the 64th order enstrophy series was evaluated. The analysis of the enstrophy singularities using Padé and quadratic approximants is discussed.     

Fixed stream-tube method for solving two-phase plane flow problems and its theoretical analysis

The fixed stream-tube method widely adopted in engineering field for giving an approximate solution to the two-dimensional problems of two-phase flow through porous media is summarized and an improvement has been made in this paper. Its core part, i.e., the fluid displacement within a one-dimensional stream tube with variable cross-sectional area under a given pressure difference across the tube is thoroughly studied. The existence and uniqueness of solution are proved, the exact solution, numerical solution and its convergence, stability analyses are given in this paper.     

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.     

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

Incomplete blowup of solutions of quasilinear hyperbolic balance laws

We study incomplete blowup of entropy solutions to first-order quasilinear hyperbolic balance laws. A general procedure to continue solutions beyond the blowup time, which makes use of monotonicity methods, is given. The continuations thus obtained are possibly unbounded and satisfy suitable generalized entropy and Rankine-Hugoniot conditions. Then the uniqueness of continuations satisfying such conditions is proved.     

Numerical Solution of the Polymer System by Front Tracking

The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is well known that the front tracking method may introduce a blowup of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time, and that this time decreases with the discretization parameter.For multidimensional problems, front tracking is combined with dimensional splitting, and numerical experiments indicate that large splitting steps can be used without loss of accuracy. Typical CFL numbers are in the range 10–20, and comparisons with Riemann free, high-resolution methods confirm the high efficiency of front tracking.The polymer system, coupled with an elliptic pressure equation, models two-phase, three-component polymer flooding in an oil reservoir. Two examples are presented, where this model is solved by a sequential time stepping procedure. Because of the approximate Riemann solver, the method is non-conservative and CFL numbers must be chosen only moderately larger than unity to avoid substantial material balance errors generated in near-well regions after water breakthrough. Moreover, it is demonstrated that dimensional splitting may introduce severe grid orientation effects for unstable displacements that are accentuated for decreasing discretization parameters.     

Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

Local Well-Posedness and Blowup Criterion of the Boussinesq Equations in Critical Besov Spaces

We consider the existence of the 2D inviscid Boussinesq equations in critical Besov spaces and obtain some blowup criteria.     

Nonlinear effects in absolute and convective instabilities of a near-wake

Linear and nonlinear initial-value problems are discussed for planar inviscid disturbances in streamlined near-wakes. This is mostly for those areas of near-wake flow where the basic motion comprises nearly uniform shear with or without normal influx into the accompanying viscous interfacial layer, although agreement is found with linear properties for full velocity profiles of double-Blasius, double-Jobe–Burggraf, Hakkinen–Rott and Goldstein form. With nonlinear disturbances, wavelike initial conditions yield a known critical-layer development, whereas more general, non-wave, initial conditions lead to a new integro-partial-differential amplitude equation which is studied analytically and numerically. The solutions show decay, finite-time blowup or nonlinear upstream-travelling disturbances. The normal influx proves crucial. Absolute and upstream- or downstream-convective instability is encountered (depending on the profiles, and flow reversal, for example); and in generic cases (for any thin airfoil) nonlinearity is shown analytically to provoke upstream convection. Increased nonlinearity drives the typical transition point extremely close to the trailing edge. Comparisons are made with three-dimensional behaviour in the linear case and with a direct simulation in the nonlinear regime.     

Approximate sampling theorem for bivariate continuous function

An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .     

BOUNDARY CONTROL OF MKDV-BURGERS EQUATION

The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1].The existence of the solution of MKdV-Burgers equation with the feedback control law was proved.On the base,priori estimates for the solution was given.At last,the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTA-TION (Ⅰ)-THE CONTINUOUS-TIME CASE

IntroductionTheshallowwaterequationsareanimportantmathematicalmodelforavarietyofprobleminhydraulicengineering .Inrecentyears,therehasbeeninterestinthenumericalsolutionfortheshallowwaterequations.Thenumericalsimulationsfortheshallowwaterequationsystemcanbeappliedtomanypurposes .First,itcanserveasameansformodelingtidalfluctuationsforthosenterestedincapturingtidalenergyforcommercialpurposes.Secondly ,thesesimulationscanbeusedtocomputetidalrangesandsurgessuchashurricanesandtsunamiscausedbyextreme…     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅰ)-THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.     

弹性力学混合状态方程的小波解法

应用小波理论求解弹性力学混合状态方程，讨论了解的收敛性。从文中的数值算例不难看出，该方法不失为混合状态方程一种新的求解途径。