Global Existence Of Classical Solutions Research Articles

Stability of the Equilibrium to the Vlasov-Poisson-Boltzmann System with Non-constant Background Charge

We study the global existence of classical solution to the Vlasov-Poisson-Boltzmann system with non-constant background charge. In this case the local Maxwellian is the unique stationary state. We show that this equilibrium is nonlinear stable provided that the initial perturbation is sufficient small. Our result solves an open problem stated by Duan and Yang (SIAM J. Math. Anal. 41(6):2353–2387, 2009) in one dimensional case.

Global existence of large solutions to conservation laws with nonlocal dissipation-type terms

We consider the global existence for large perturbation solutions of scalar conservation laws with nonlocal dissipative structures.At the very beginning, based on the Green function method and the Littlewood-Paley decomposition, we establish a new regularity criterion to improve the regularity of the solution. It is worth to point out that the key points in the regularity criterion are that $\|u\|_{L^\infty}<+\infty$ for $0\leq~s<1$ and $\|u\|_{C^\alpha}<+\infty$ for $s=1$. Thus, in the following part we devote ourselves to verify the two key points. For the subcritical case, we obtain the boundedness of solution in $L^p$ for any $p\in[2,~+\infty)$ by the maximum principle. Thanks to the nonlinear maximum principle, $C^\alpha$ boundedness is established for the critical case.Finally, the global existence of classical solutions to the scalar conservation with large initial data is obtained.

Global well-posedness of the 3D non-isothermal compressible fluid model of Korteweg type

The aim of this work is to study the global existence and large-time behavior of solutions to the non-isothermal model of capillary compressible fluids derived by Dunn and Serrin (1985). It is proved that the three-dimensional non-isothermal compressible Navier–Stokes–Korteweg system admits a unique global classical solution, provided that the initial energy is suitably small. This result improves previous results obtained by Hattori and Li (1996), where the existence of global classical solutions is established for the initial data close to an equilibrium in some Sobolev space Hs(s≥3). Furthermore, the large time behavior of the solution is also investigated.

Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization

The interplay of chemotaxis, convection and reaction terms is studied in the particular framework of a refined model for coral broadcast spawning, consisting of three equations describing the population densities of unfertilized sperms and eggs and the concentration of a chemical released by the latter, coupled to the incompressible Navier–Stokes equations.Under mild assumptions on the initial data, global existence of classical solutions to an associated initial-boundary value problem in bounded planar domains is established. Moreover, all these solutions are shown to approach a spatially homogeneous equilibrium in the large time limit.

Singularity Formation and Global Existence of Classical Solutions for One-Dimensional Rotating Shallow Water System

We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of ${C}^1$ initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.

Global Existence of Inhomogeneous Incompressible Isotropic Elastodynamics in Three Dimensions

This paper concerns the Cauchy problem of the incompressible isotropic elastodynamics in three dimensions. The homogeneous case was studied by Sideris and Thomases [Comm. Pure Appl. Math., 58 (2005), pp. 750--788; \it Comm. Pure Appl. Math., 60 (2007), pp. 1707--1730] in Euler coordinates. Here we generalize their result to the inhomogeneous case and prove the global existence of classical solution in Lagrangian coordinates under the assumption that the density is a small perturbation around a constant state. The generalized energy method is utilized to derive the energy estimates in certain weighted Sobolev spaces.

Global classical solutions to reaction-diffusion systems in one and two dimensions

The global existence of classical solutions to reaction-diffusion systems in dimensions one and two is proved. The considered systems are assumed to satisfy an {\it entropy inequality} and have nonlinearities with at most cubic growth in 1D or at most quadratic growth in 2D. This global existence was already proved in [T. Goudon and A. Vasseur, Ann. Sci. Ecole Norm. Sup. (4) 43 (2010), no. 1, 117--142] by a De Giorgi method. In this paper, we give a simplified proof by using a modified Gagliardo-Nirenberg inequality and the regularity of the heat operator. Moreover, the classical solution is proved to have $L^{\infty}$-norm growing at most polynomially in time. As an application, solutions to chemical reaction-diffusion systems satisfying the so-called complex balance condition are proved to converge exponentially to equilibrium in $L^{\infty}$-norm.

Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations

Global existence of classical solutions to the relativistic Vlasov-Maxwell system, given sufficiently regular initial data, is a long-standing open problem. The aim of this project is to present in details the results of a paper published in 1986 by Robert Glassey and Walter Strauss. In that paper, a sufficient condition for the global existence of a smooth solution to the relativistic Vlasov-Maxwell system is derived. In the following, the resulting theorem is proved by taking initial data , . A small data global existence result is presented as well.

Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity $\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &amp;x∈ Ω, ~~t&gt;0, \\ c_t+u·\nabla c = Δ c+ n-c,&amp;x∈ Ω, ~~t&gt;0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&amp;x∈ Ω, ~~t&gt;0, \\\nabla · u = 0&amp; x∈ Ω, \ t&gt;0, \end{cases}$ in a bounded smooth domain $Ω\subset \mathbb{R}^2$ with homogeneous Neumann boundary conditions, where $a≥0$ and $b&gt;0$ are constants, and the functions $d(c)$ and $χ(c)$ satisfy the following assumptions:● $(d(c), χ (c))∈ [C^2([0, ∞))]^2$ with $d(c), χ(c)&gt;0$ for all $c≥0$, $d'(c)&lt;0$ and $\lim\limits_{c\to∞}d(c) = 0$.● $\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$ and $\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$ exist.The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition $\lim\limits_{c\to∞}d(c) = 0$. In this paper, we will use function $d(c)$ as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution $(n, c, u)$ will converge to the constant state $(\frac{a}{b}, \frac{a}{b}, 0)$ if $b&gt;\frac{K_0}{16}$ with $K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$.

On the global solvability of the coupled kinetic-fluid system for flocking with large initial data

We study the dynamics of infinitely many Cucker–Smale (CS) flocking particles under the interplay of random communication and compressible fluids in planar wave case. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the compressible Navier–Stokes (NS) equations. These two subsystems are coupled via the drag force. For this coupled model, we present a global existence of classical solutions for arbitrarily large initial data which may contain vacuum.

Classical solutions for the Vlasov–Poisson system with damping term

ABSTRACTWe investigate global existence of classical solution to the three dimensional Vlasov–Poisson system with radiation damping . We prove that a small perturbation of initial datum for a global classical solution verifying certain decay conditions also launches a global solution which has sharp decay. In addition, we prove that a quasineutral initial datum launches a global classical solution.

Global solvability and explicit bounds for non-local adhesion models

Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.

Global Classical Solutions of Three Dimensional Viscous MHD System Without Magnetic Diffusion on Periodic Boxes

In this paper, we study the global existence of classical solutions to the three dimensional incompressible viscous magneto-hydrodynamical system without magnetic diffusion on periodic boxes, i.e., with periodic boundary conditions. We work in Eulerian coordinate and employ a time-weighted energy estimate to prove the global existence result, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have some symmetries.

Global existence of classical solutions to a cross-diffusion predator–prey system with a free boundary

This paper is concerned with a cross-diffusion predator–prey system with a free boundary over a one-dimensional habitat. The free boundary shows the spreading front of the prey and predator which implies that the velocity of the expanding front is proportional to the gradients of the prey and predator. By the contraction mapping principle, $$L^{p}$$ estimates and Schauder estimates of parabolic equations, the local and global existence and uniqueness of classical solutions are established for this system.

A Sixth-Order Phase-Field Equation with Degenerate Mobility

In this paper, we study the weak solutions of a sixth-order phase-field equation with degenerate phase-dependent diffusion mobility in 2D case. The main features and difficulties of this equation are given by a highly nonlinear sixth-order elliptic term, a strong constraint imposed by the presence of the nonlinear principal part and the lack of maximum principle. Based on the Schauder-type estimates and entropy estimates, we are able to prove 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 by using approximation and compactness tools. In the end, we study the nonnegativity of solutions for the sixth-order degenerate phase-field equation.

Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity

The Neumann initial-boundary value problem for the chemotaxis system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u,(⋆)is considered in a bounded domain , , with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D(s) decays exponentially as . In such situations, namely, straightforward Moser-type recursive arguments for the derivation of estimates for u from corresponding Lp bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context.This work develops an alternative approach, at its core based on a Moser-type iteration for the quantity , to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of as , under the assumptions that with some K1 > 0, K2 > 0, and we have for all , and that the size of S relative to D can be estimated according to for all with some K3 > 0 and .Making use of the fact that this allows for certain superalgebraic growth of , as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when and for all and some positive α and β, the assumptions that and that0\\quad \\quad \ ext{and}\\quad \\quad \\left\\{\\begin{array}{*{35}{l}} \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right)\\quad \\quad & \ ext{if}~n=2, \\\\ \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right]\\quad \\quad & \ ext{if}~n\\geqslant 3, \\end{array}\\right. \\end{eqnarray} ?>β>0and{α∈(β2,β)if n=2,α∈(β2,β]if n⩾3,warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic.To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller–Segel system of type () for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions.

Global existence and asymptotic behavior for the full compressible Euler equations with damping in $\mathbb R^3$

We are concerned with the global existence and asymptotic behavior of classical solutions to the Cauchy problem for the full compressible Euler equations with damping in $\mathbb R^3$. We prove the global existence of the classical solutions by the delica

Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods

The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's results on global existence of classical solutions still hold under much weaker condition that their BMO norms do not blow up in finite time. The proof makes use of some new global and local weighted Gagliardo-Nirenberg inequalities involving BMO norms.

Global solutions for a hyperbolic–parabolic system of chemotaxis

We study a hyperbolic–parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two. We consider diffusions given by the fractional Laplacian and, in particular, we prove the global existence of classical solutions in certain dissipation regimes.

Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L ∞ estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [ 4 , 30 ]) require that the density is bounded from below by a positive constant.