Global Existence Of Classical Solutions Research Articles

The plasma-charge model in a convex domain

The aim of this paper is to study the initial-boundary value problems of a Vlasov type system in a convex domain, so called the plasma-charge model, in which there are two kinds of singular sets, one caused by the boundary effect, the other by the heavy point charges. We prove the local existence of classical solutions for the case that the point charges are moving and global existence of classical solutions for the case that the point charges are fixed away from the boundary. The crucial tools are the extended Velocity Lemma for the plasma-charge model and the Pfaffelmoser’s method developed by Hwang and Velázquez (2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26). In the Pfaffelmoser’s argument, a new idea is that the plasma particles can only be close to one of the singular sets during the time interval [t−δ,t] with small length δ, which allows us to obtain the global existence for the fixed point charges case by adapting the techniques established by Hwang et al (2013 Discrete Contin. Dyn. Syst. 33 723–37; 2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26) to the corresponding singular sets respectively.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

Global dynamics and spatiotemporal heterogeneity of a preytaxis model with prey-induced acceleration

Abstract Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.

Global well‐posedness and long‐time behavior in a tumor invasion model with cross‐diffusion

AbstractThis paper is concerned with a cross‐diffusion tumor invasion model with double‐taxis effect. We first investigate the global existence of classical solutions of this model in two‐dimensional space. The essential difficulty lies in the second‐level taxis effect of immune cells on tumor cells, where chemotactic factor (tumor cells) exhibit their own taxis behavior, the double‐taxis effect makes us have to use more detailed analysis and calculation, and some new estimation techniques are used. Subsequently, we also investigate the stability of some equilibria. For small proliferation coefficient, we prove the global asymptotic stability or local asymptotic stability of a semitrivial equilibrium point. While, for the other equilibria, the stability analysis is complicated even for some special cases, and both the double chemotactic coefficients will affect the stability of the solution.

Existence of Classical Solutions for Broer-Kaup Equations

In this paper we investigate the Cauchy problem for one dimensional Broer-Kaup equations for existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.

Global Existence of Classical Solutions to an Aggregation Model with Logistic Source

Global Existence of Classical Solutions to an Aggregation Model with Logistic Source

Global classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition in 2D bounded domains

We study the barotropic compressible Navier–Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with C∞ boundary ∂Ω. By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Furthermore, we also prove that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum (even a point) appears initially. As we known, this is the first result concerning the global existence of classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition and the density containing vacuum initially for general 2D bounded smooth domains.

Global boundedness in an attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanism

We study the following chemotaxis system involving nonlinear indirect signal mechanism{ut=Δu−ξ∇⋅(u∇v)+χ∇⋅(u∇z),x∈Ω,t>0,vt=Δv−v+wγ1,x∈Ω,t>0,0=Δw−w+uγ2,x∈Ω,t>0,0=Δz−z+uγ3,x∈Ω, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn(n≥1), where ξ,χ,γ1,γ2,γ3>0. With the aid of maximum Sobolev regularity and a priori estimates of w and z, it has been proved that if either random diffusion or repulsion mechanism dominates the nonlinear indirect attraction mechanism with γ1γ2<max⁡{2n,γ3}, then the system admits a global classical solution, which is bounded in Ω×(0,∞). Moreover, we also show that under balance situations with γ1γ2=max⁡{2n,γ3}, the existence of global classical solution is determined by the sizes of chemotaxis sensitivity coefficients ξ and χ as well as the initial data u0. This work extends the previous results, where the production of signal is nonlinear mechanism.

Existence of Classical Solutions for a Class of the Khokhlov-Zabolotskaya-Kuznetsov Type Equations

In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov--Zabolotskaya--Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of~the~Khokhlov--Zabolotskaya--Kuznetsov type equations for the existence of global classical solutions. We~give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.

Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems

Abstract We study the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction. In both L 1 {L^{1}} and L ∞ {L^{\infty}} frameworks, we will first show the global existence of classical solutions with small initial data, and then prove that the global solution will scatter to free linear waves and study the rigidity aspect of the scattering problem. We also show the inverse scattering result: The scattering field can determine the global solution uniquely, in the L 1 {L^{1}} framework.

Global existence of classical solutions and numerical simulations of a cancer invasion model

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions.

Global solvability in a two-dimensional singular chemotaxis-fluid system with indirect signal production or consumption

This paper deals with the singular chemotaxis-fluid system with indirect signal production or consumption nt+u⋅∇n=Δn−χ∇⋅(nv∇v); vt+u⋅∇v=Δv+g(v,w); wt+u⋅∇w=Δw−w+n; ut+(u⋅∇)u=Δu−∇P+n∇Φ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R2 with no-flux/no-flux/no-flux/Dirichlet boundary conditions, where Φ∈W2,∞(Ω). In the cases signal production (i.e., g(v,w)=−v+w) and consumption (i.e., g(v,w)=−vw), it is proved that for any appropriately regular initial data, the associated initial-boundary value problem possesses a global classical solution for any χ>0. It is worth mentioning that the global existence of classical solution for χ∈(0,1) or generalized solutions for arbitrary χ>0 was obtained in the previously known results on the corresponding system with direct signal production, and that only the global generalized solutions for arbitrary χ>0 were established in a precedent concerning the corresponding system with direct signal consumption. In comparison to these results for the case of direct signal production or consumption, our results rigorously confirm that both the indirect signal production and consumption mechanisms authentically contribute to the global solvability of the two-dimensional singular chemotaxis-fluid system.

Global existence of classical solutions to a regularized brine inclusion model

Global existence of classical solutions to a regularized brine inclusion model

Global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time

In this study, we examine the global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time. It is described by the initial-boundary value problem of compressible Euler equations together with a sufficiently small variable parameter, which can be viewed as a hyperbolic system of balance laws whose source is a non-dissipative term when Riemann invariants are applied. We prove a couple of global existence theorems of classical solutions under the suitable conditions of expanding ducts and the initial-boundary data whose C0 norms can be large. The analysis depends primarily on the local existence theorem and on uniform a priori estimates, which are obtained by giving the maximum principle and introducing new generalized Lax transformations. Furthermore, the limit behavior of expanding ducts at infinity and the long-time behavior of global classical solutions along all characteristic curves and vertical lines are also determined. Lastly, we explore the feasibility of the initial value problem for such expanding ducts.

Global classical solutions to the viscous two-phase flow model with slip boundary conditions in 3D exterior domains

We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.

Global Well-Posedness and Large-Time Behavior for the Equilibrium Diffusion Model in Radiation Hydrodynamics

In this paper, we investigate the global well-posedness of the equilibrium diffusion model in radiation hydrodynamics. The model consists of the compressible Navier-Stokes equations coupled with radiation effect terms described by the fourth power of temperature. The global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state: moreover, an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the refined energy method and Fourier’s analysis.

Stochastic Lagrangian perturbation of Lie transport and applications to fluids

In this paper, we propose a novel stochastic Lagrangian formulation of dissipatively perturbed Lie transport, which is based on the statistical generalized Cauchy invariants equation. This formulation consists of, first, finding a convenient Lagrangian formulation of the Lie transport equation involving particle trajectories, for instance the backward generalized Cauchy invariants equation (a Lagrangian formulation for Lie-advected exact p-forms, which is the Hodge dual of a generalization of the Cauchy vorticity formula), and second, performing a stochastic perturbation of the velocity-driven particle trajectories by adding to them white noises. Finally, using Itô’s calculus, an ensemble average of the stochastically perturbated generalized Cauchy invariants equation allows us to obtain the Lie transport equation perturbed by a deterministic potentially dissipative term given by the sum of squares of some Lie derivative operators. A remarkable property of this equation is that it satisfies a statistical Kelvin–Helmholtz theorem of conservation of circulation and flux. These results are obtained on periodic Euclidean spaces as well as on smooth closed Riemannian manifolds. In particular, we recover and thus generalize the Constantin–Iyer results on the stochastic Lagrangian formulation of the incompressible Navier–Stokes equations to a larger class of (deterministic) dissipative PDEs. A first application of this stochastic Lagrangian formulation is the derivation of new Lagrangian formulations for non-ideal (dissipative) hydrodynamic and magnetohydrodynamic models on flat and curved spaces, and in particular we obtain stochastic-Lagrangian incompressible extended MHD equations. As a second application, we use this new stochastic Lagrangian formulation to study the local well-posedness, the non-resistive limit and the global existence of classical solutions for the non-ideal incompressible extended MHD on the flat torus. Global-in-time existence is proved for small magnetic Reynolds numbers, that is, either for small initial data or large resistivity.

Global existence and uniform boundedness in a fully parabolic Keller–Segel system with non-monotonic signal-dependent motility

This paper is concerned with global solvability of a fully parabolic system of Keller–Segel-type involving non-monotonic signal-dependent motility. First, we prove global existence of classical solutions to our problem with generic positive motility function under a certain smallness assumption at infinity, which however permits the motility function to be arbitrarily large within a finite region. Then uniform-in-time boundedness of classical solutions is established whenever the motility function has strictly positive lower and upper bounds in any dimension N≥1, or decays at a certain slow rate at infinity for N≥2. Our results remove the crucial non-increasing requirement on the motility function in some recent work [10,13,16] and hence allow for both chemo-attractive and chemo-repulsive effect, or their co-existence in applications. The key ingredient of our proof lies in an important improvement of the comparison method developed in [10,16,18].

Global existence of a nonlinear wave equation arising from Nordström’s theory of gravitation

We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus {mathord {mathbb T}}^3 in the Sobolev spaces H^m({mathord {mathbb T}}^3). We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up result if the conditions of our global existence theorems are not met.

Global existence for quasilinear wave equations exterior to star-shaped regions with inhomogeneous boundary conditions

This paper establishes the global existence of classical solutions to systems of nonlinear wave equations with multiple speeds outside star-shaped regions satisfying time-independent inhomogeneous boundary conditions, provided that nonlinear terms obey the null condition. We first prove the existence and uniqueness of stationary solutions. Then we use the space-time estimates for perturbed wave equations in Metcalfe and Sogge (2007) to show that solutions of systems converge to stationary solutions as time goes to infinity.