Local Classical Solution Research Articles

Global Solution and Stability of a Haptotaxis Mathematical Model for Complex MAP

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system () has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system () from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u*,v*,w*) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (), that is, it can have a destabilizing effect.

Noble-Abel gas diffusion at convex corners of the two-dimensional compressible magnetohydrodynamic system

<p>In this paper, we study the expansion of Noble-Abel gas into a vacuum around the convex corner of the two-dimensional compressible magnetohydrodynamic system. We reduce this problem to the interaction of a centered simple wave emanating from the convex corner with a backward planar simple wave. Mathematically, this is a Goursat problem. By using the method of characteristic decomposition and construction of invariant regions, combining $ C^{0} $ and $ C^{1} $ estimation as well as hyperbolicity estimation, we obtain the existence of a global classical solution by extending the local classical solution.</p>

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.

Global existence and decay estimates to a coupled Cahn-Hilliard system on [formula omitted

In this paper, we study the Cauchy problem of a coupled Cahn-Hilliard system. First, we construct the local classical solution, then by some a priori estimates, we extend the local solution step by step to all t>0 provided that ‖ϕ0‖W1,1(RN)+‖ψ0‖L1(RN) sufficiently small. Finally, based on the Fourier splitting method and the time-independent L1-a priori estimate, we show the temporal decay estimates of the global solution.

The degenerate hyperbolic problem for the two‐dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.

Continuum limit of the lattice Lohe group model and emergent dynamics

We study the emergent dynamics and global well‐posedness of the matrix‐valued integro‐differential equation which can be derived from the continuum limit of the lattice Lohe group model. The lattice Lohe group model corresponds to the generalized high‐dimensional Kuramoto model. The solution to the lattice Lohe group model can be cast as a simple function‐valued solution to the continuum Lohe group model. We first construct a local classical solution to the continuum Lohe group model, and then we find an invariant set and derive a global well‐posedness in some sufficient frameworks formulated in terms of initial data, system functions, and system parameters. We also show that phase‐locked states can emerge from the admissible class of initial data in a large coupling regime. Moreover, we show that sequence of simple functions obtained from the solutions of the lattice Lohe group model converges to a local classical solution to the continuum Lohe group model in supremum norm.

Zero-electron-mass limit of the compressible Navier–Stokes–Poisson equations with well/ill-prepared initial data

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

Sonic-supersonic Solutions to a Mixed-type Boundary Value Problem for the Two-Dimensional Full Euler Equations

We are concerned with the sonic-supersonic structures extracted from the transonic flow problems in gas dynamics. A local classical supersonic solution for the two-dimensional steady full Euler equations is established in an angular region bounded by the sonic curve and the characteristic curve. In order to overcome the challenges caused by the coupling of nonlinearity and degeneracy at the corner point, we develop a new iteration pattern to show the convergence of the iterative sequence generated by the Euler equations in terms of a partial hodograph coordinate system. The pattern developed here will be useful for studying the degenerate mixed-type boundary value problems for other related nonlinear hyperbolic systems.

带有非局部效应耦合流体模型的局部经典解

带有非局部效应耦合流体模型的局部经典解

On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry

AbstractThe transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data given on the degenerate hyperbolic curve. By introducing a novel set of dependent and independent variables, we use the idea of characteristic decomposition to transform the axisymmetric Euler equations as a new system which has explicitly singularity-regularity structures. We first establish a local classical solution for the new system in a weighted metric space and then convert the solution in terms of the original variables.

Global classical solutions to quadratic systems with mass control in arbitrary dimensions

The global existence of classical solutions to reaction-diffusion systems in arbitrary space dimensions is studied. The nonlinearities are assumed to be quasi-positive, to have (slightly super-) quadratic growth, and to possess a mass control, which includes the important cases of mass conservation and mass dissipation. Under these assumptions, the local classical solution is shown to be global, and in the case of mass conservation or mass dissipation, to have the L∞-norm growing at most polynomially in time. Applications include skew-symmetric Lotka-Volterra systems and quadratic reversible chemical reactions.

On a degenerate mixed-type boundary value problem to the 2-D steady Euler equation

This paper is concerned with the structure of sonic-supersonic solutions near sonic curves arising from the transonic channel flow problems in gas dynamics. Given two smooth curves, one of which is a sonic curve and the other is a characteristic curve, we construct a local classical supersonic solution for the two-dimensional steady Euler equations in the angular region. A novel set of dependent and independent variables are introduced to transform the Euler equations into a linear system with explicitly singularity-regularity structures. We use the iteration method to establish the existence and uniqueness of smooth solutions for the new system and then express the solution in terms of the original variables.

Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations

Through applying Galerkin method, we establish the approximating solution for one-dimension Klein-Gordon-Zakharov equations and obtain the local classical solution. By applying integral estimates, we also obtain the existence and uniqueness of the global classical solution of Klein-Gordon-Zakharov equations.

Existence of the solution for the viscous bipolar quantum hydrodynamic model

In this paper, we investigate the existence of classical solution of the viscous bipolar quantum hydrodynamic(QHD) models for ir-rotational fluid in a periodic domain. By applying the iteration method, we prove that the viscous bipolar QHD model has a local classical solution. Then we prove this solution is global with small initial data, based on a series of a priori estimates. Finally, we obtained the inviscid limit of this viscous quantum hydrodynamic model.

Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum

This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by deriving the smoothing effect of the strong solution for $t>0$.

Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles

We use an energetic variational approach to derive a new hydrodynamic model, which could be called a generalized Poisson–Nernst–Planck–Navier–Stokes system. Such the system could describe the dynamics of the compressible conductive fluid with the dilute charged particles and be used to analyze the interactions between the macroscopic fluid motion and the microscopic charge transportation. Then, we develop a general method to obtain the unique local classical solution, the unique global solution under small perturbations and the optimal decay rates of the solution and its derivatives of any order.

On the existence of local classical solution for a class of one-dimensional compressible non-newtonian fluids

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.

Global Existence of Solutions of Cauchy Problem for Generalized Bbm-Burgers Equation

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equation (1)ut−αΔut − βΔu + γΔ2u+∑j=1nfj(u) xj = Δg(u)+G(u), x∈ Rn,t>0 are proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation (1) are proved when n=3,2,1. Moreover, the decay property of the solution is discussed.

Incompressible Limit of Viscous Liquid-Gas Two-Phase Flow Model

We investigate the incompressible limit of the viscous liquid-gas two-phase flow model with periodic boundary conditions. It is shown that the local classical solution of the two-phase flow model converges to the local classical solution of the incompressible Navier--Stokes equations, as $\epsilon\rightarrow \infty$, which builds up the relationship between the two-phase flow model and the incompressible Navier--Stokes equations. In addition, we give the convergence rate estimates in some norms.