Global Existence Of Classical Solutions Research Articles

Long-time behavior for the Hele–Shaw–Cahn–Hilliard system

We study the Hele–Shaw–Cahn–Hilliard system that models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. In the 3D case, we prove eventual regularity of weak solutions, as well as existence of global classical solutions if either the Peclet number is sufficiently small or the initial datum is close to one local energy minimizer of the free energy. In both 2D and 3D, we demonstrate that the ω-limit set of each trajectory consists of a single steady state. Finally, stability of local minimizers is established.

Global Spherically Symmetric Classical Solution to Compressible Navier–Stokes Equations with Large Initial Data and Vacuum

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier–Stokes equations with vacuum in a bounded domain or exterior domain $\Omega$ of $\mathbb{R}^n$($n\ge2$). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1–28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91–129; S.J. Ding, H.Y. Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696–1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension.

Global Existence of Classical Solutions to a Three‐Species Predator‐Prey Model with Two Prey‐Taxes

We are concerned with three‐species predator‐prey model including two prey‐taxes and Holling type II functional response under no flux boundary condition. By applying the contraction mapping principle, the parabolic Schauder estimates, and parabolic Lp estimates, we prove that there exists a unique global classical solution of this system.

The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.

Global solutions to the relativistic Landau–Maxwell system in the whole space

The relativistic Landau–Maxwell system is one of the most fundamental models for description of the dynamics of dilute cold plasma in which particles interact through the Coulomb collision in the self-consistent electro-magnetic field. By constructing the compensating functions to this system and by using the structure of the equations, we obtain the global existence of classical solutions to this system in the whole space. For a simpler model, that is, the relativistic Landau–Poisson system, the analysis yields the optimal convergence rate in time to the equilibrium.

Global classical solutions of coagulation–fragmentation equations with unbounded coagulation rates

We study discrete fragmentation coagulation equations in spaces X p , p > 1 , consisting of distributions having the p th moments finite. We show that for sufficiently regular fragmentation laws the fragmentation semigroup is analytic in X p , and fully characterize the domain of its generator. This allows for explicit characterization of the domains of the fractional powers of the generator through real interpolation. Finally, we use the linear results to show the existence of global classical solutions to fragmentation coagulation equations for a class of unbounded coagulation kernels.

Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimension

We establish the existence of global classical solutions and non-trivial steady states of a one-dimensional attraction–repulsion chemotaxis model subject to the Neumann boundary conditions. The results are derived based on the method of energy estimates and the phase plane analysis.

Global solutions of initial problems for hyperbolic functional differential systems

A generalized Cauchy problem for almost linear hyperbolic func- tional differential systems is considered. A theorem on the global existence of classical solutions is proved. It is shown a result on the differentiability of solu- tions with respect to initial functions. A method of characteristics and integral functional inequalities are used.

EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY-PREDATOR MODEL WITH NON-MONOTONIC FUNCTIONAL RESPONSE AND CROSS-DIFFUSION

In this paper, using the energy estimates and the bootstrap arguments, the global existence of classical solutions for a prey-predator model with non-monotonic functional response and cross-diffusion where the prey and predator both have linear density restriction is proved when the space dimension n

Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system

We are concerned with the global well-posedness of a two-phaseflow system arising in the modelling of fluid-particleinteractions. This system consists of the Vlasov-Fokker-Planckequation for the dispersed phase (particles) coupled to theincompressible Euler equations for a dense phase (fluid) throughthe friction forcing. Global existence of classical solutions tothe Cauchy problem in the whole space is established when initialdata is a small smooth perturbation of a constant equilibriumstate, and moreover an algebraic rate of convergence of solutionstoward equilibrium is obtained under additional conditions oninitial data. The proof is based on the macro-micro decompositionand Kawashima's hyperbolic-parabolic dissipation argument. Thisresult is generalized to the periodic case, when particles are inthe torus, improving the rate of convergence to exponential.

Global existence and uniqueness of a three-dimensional model of cellular electrophysiology

We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of quasipositivity principle and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.

QUALITATIVE PROPERTIES FOR A SIXTH–ORDER THIN FILM EQUATION

In this article, the author studies the qualitative properties of weak solutions for a sixth‐order thin film equation, which arises in the industrial application of the isolation oxidation of silicon. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the expansion of the support of solutions are also discussed.

On the classical solutions of two dimensional inviscid rotating shallow water system

We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid rotating shallow water system with small initial data subject to the zero relative vorticity condition. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein–Gordon system with quadratic nonlinearity, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal form method. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity.

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation-strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under the smallness assumptions only on the size of initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.

Global Existence of Classical Solutions to the Boltzmann Equation with External Force for Hard Potentials

The Boltzmann equation with external force describes the time evolution of rarefied gas in an external field. In this paper, we construct the global existence of classical solutions to the Boltzmann equation for hard potentials with Grad's angular cutoff in the whole space when the initial datum is a perturbation of the local Maxwellian. The proof relies on the nonlinear energy method where the crucial step is to show the positivity of the linearized collision operator. Further, we introduce new energy functional to obtain the desired differential inequality.

Weak and classical solutions to predator–prey system with cross-diffusion

In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator–prey system with cross-diffusion in heterogeneous habitats. Predators are assumed to feed on preys with a Holling type II functional response to prey density and preys are assumed to follow a logistic growth in the absence of predation. The mobility of each classes is assumed to be influenced by the gradient of other classes. The existence result is proved by means of an approximation system, the Faedo–Galerkin method, and the compactness method. The global existence of classical solutions is proved under certain restrictions on the coefficients.

Global classical solutions to hyperbolic geometric flow on Riemann surfaces

We consider the Cauchy problem for hyperbolic geometric flow equations introduced recently by Kong and Liu motivated by the Einstein equation and Hamilton Ricci flow, and obtain a necessary and sufficient condition for the global existence of classical solutions to this kind of flow on Riemann surfaces. The results show that the scalar curvature of the solution metric g ij converges to one of flat curvature, and the hyperbolic geometric flow has the advantage that the surgery technique may be replaced by choosing a suitable initial velocity tensor.

Classical solutions to the Vlasov–Poisson system in an accelerating cosmological setting

In this paper, we study the Vlasov–Poisson system in an accelerating cosmological setting. A homogeneous and isotropic background solution, which describes an expanding universe in an acceleration, is constructed explicitly, and then its perturbed solutions are investigated to obtain global existence of classical solutions and the uniform L 1 -stability estimates.

Quasi-steady-state approximation for a reaction–diffusion system with fast intermediate

We consider a prototype reaction–diffusion system which models a network of two consecutive reactions in which chemical components A and B form an intermediate C which decays into two products P and Q. Such a situation often occurs in applications and in the typical case when the intermediate is highly reactive, the species C is eliminated from the system by means of a quasi-steady-state approximation. In this paper, we prove the convergence of the solutions in L 2 , as the decay rate of the intermediate tends to infinity, for all bounded initial data, even in the case of initial boundary layers. The limiting system is indeed the one which results from formal application of the QSSA. The proof combines the recent L 2 -approach to reaction–diffusion systems having at most quadratic reaction terms, with local L ∞ -bounds which are independent of the decay rate of the intermediate. We also prove existence of global classical solutions to the initial system.