Local 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.

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 $t=0$.

Local Existence of Classical Solutions to Scalar Field Equation on Spatially Compact Spacetime as a Background

The goal of this paper is to prove the wellposedness of scalar field equation on spatially compact spacetime of Riemannian manifold. We construct the equation of motion from the Lagrangian of scalar field with non-minimal coupling, where the coupling interaction of the scalar field ϕ is proportional to the scalar curvature of the spacetime. The equation of motion has the form like non-linear wave equation. The next step is to prove local existence of solutions. We have show that both the k th linear energy and energy norm are bounded for the finite time with the initial data in H k+1 × H k. Finally, we prove the uniqueness and smoothness properties of the solution.

On the global existence for the compressible Euler–Poisson system, and the instability of static solutions

We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index $$\gamma $$ of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Ser I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field $$u_0$$ such that the spectrum of $$Du_0$$ is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

On the existence of local classical solutions to the Navier-Stokes equations with degenerate viscosities

In this paper, we consider the three-dimensional Cauchy problem of compressible Navier-Stokes equations with degenerate viscosities depending on density. With the initial data containing vacuum at far fields, we prove the local existence of regular solutions with some compatibility conditions. The regular solution is classical due to the smoothing effect of the velocity u.

On a degenerate boundary value problem to the two-dimensional self-similar nonlinear wave system

This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic degeneracy, we introduce a partial hodograph transformation to transform the nonlinear wave system into a new system, which displays a clear regularity–singularity structure. The local existence of classical solutions for the new system is established in a weighted metric space. Returning the solution to the original variables, we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.

New results on IBVP for Class of Nonlinear Parabolic Equations

In this article we propose a new approach for investigation the local existence of classical solutions of IBVP for a class of nonlinear parabolic equations.

Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang on the two-dimensional Prandtl equations to the three-dimensional setting.

Boundedness in a Keller–Segel system with external signal production

We study the Neumann initial-boundary problem for the chemotaxis system{ut=Δu−∇⋅(u∇v),x∈Ω,t>0,vt=Δv−v+u+f(x,t),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω in a smooth, bounded domain Ω⊂Rn with n≥2 and f∈L∞([0,∞);Ln2+δ0(Ω))∩Cα(Ω×(0,∞)) with some α>0 and δ0∈(0,1). First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of n=2 and f being constant in time, requiring the nonnegative initial data u0 to fulfill the property ∫Ωu0dx<4π ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller–Segel model (coinciding with f≡0 in the system above) to the case f≢0. Under certain smallness conditions imposed on the initial data and f we furthermore show that for more general space dimension n≥2 and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller–Segel system to the present situation.

Global existence for a semi-linear Volterra parabolic equation and neutral system with infinite delay

This paper studies the global and local existence of classical solutions for a semilinear Volterra integro-differential equation of parabolic type: (u+k*u)′=A(u+k*u)+f(u)+g, where A is a (not necessarily densely defined) sectorial operator with its spectrum contained in the left half plane. We transform the Volterra equation into a neutral system with infinite delay assuming the history φ of the system is known. The inverse function theorem is then employed to prove the global existence of classical solution to the system for appropriate “small” data (g, φ) if 0 belongs to the resolvent set of A. An example of the linear part being non-densely defined elliptic operators is shown to illustrate the existence theorems, and an application of our results to compressible viscoelastic fluids with hereditary viscosity is also addressed.

Global existence and blowup of solutions of the heat equation with a memory boundary of polynomial type

This paper deals with the long time behavior of solutions of the heat equation with a memory boundary of polynomial type. We rst establish the comparison principle and prove the local existence of classical solutions. Then we study the global existence and blowup of the solutions by using the comparison principle and Greens function. At last, we consider the boundary blowup and the blowup rate.

Local well-posedness of the three dimensional compressible Euler–Poisson equations with physical vacuum

This paper is concerned with the three dimensional compressible Euler–Poisson equations with moving physical vacuum boundary condition. This fluid system is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local existence of classical solutions for initial data in certain weighted Sobolev spaces is established in the case that the adiabatic index satisfies 1<γ<3.

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 and blowup for a degenerate and singular parabolic system with nonlocal source and absorptions

The aim of this paper is to investigate the asymptotic behavior of positive solutions to the following degenerate and singular parabolic system $$\begin{array}{lll}{u_t}=(x^ \alpha u_x)_x+ \int \limits_{0}^{a} v^{p_1}{\rm d} x - {k_1} u^{q_1}, \quad 0 0, \\ \upsilon_{t}=(x^{\beta} v_{x})_{x} + \int \limits_{0}^{a} u^{p_2}{\rm d}x - k_{2} \upsilon^{q_2}, \, \, \, \, \, 0 0,\\ u(0,t) = u(a,t) = \upsilon(0,t) = \upsilon(a,t) = 0, \, \, \, \, \, t > 0,\\ u(x,0) = u_{0}(x) \geq 0, \upsilon(x,0) = \upsilon_{0}(x) \geq 0,\, \, \, \, \, 0 \leq x \leq a,\end{array}$$ where the constants $${0 \leq \alpha, \beta 0}$$ . Under appropriate hypotheses, we first prove a local existence of classical solution by a regularization method. Then, we discuss the global existence and blowup of positive solutions by using a comparison principle. Finally, we give the precise blowup rate estimates and the uniform blowup profiles by using the method developed by Souplet [19].

A non-local evolution equation model of cell–cell adhesion in higher dimensional space

A model for cell–cell adhesion, based on an equation originally proposed by Armstrong et al. [A continuum approach to modelling cell–cell adhesion, J. Theor. Biol. 243 (2006), pp. 98–113], is considered. The model consists of a nonlinear partial differential equation for the cell density in an N-dimensional infinite domain. It has a non-local flux term which models the component of cell motion attributable to cells having formed bonds with other nearby cells. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.

Local existence with mild regularity for the Boltzmann equation

Without Grad's angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of theangular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators.

Well-posedness for the fractional Landau–Lifshitz equation without Gilbert damping

The main purpose is to consider the well-posedness of the fractional Landau– Lifshitz equation without Gilbert damping. The local existence of classical solutions is obtained by combining Kato’s method and vanishing viscosity method, by carefully choosing the working space. Since this equation is strongly degenerate and nonlocal and no regularizing effect is available, it is a challenging problem to extend this smooth solution to global. Instead, we give some regularity criteria to show that the solution is global if some additional regularity is assumed, which seems minimal in the sense of dimensional analysis. Finally, we introduce the commutator and show the global existence of weak solutions by vanishing viscosity method.

Local Existence of Classical Solutions to Shallow Water Equations with Cauchy Data Containing Vacuum

In this paper, we investigate the Cauchy problem for the rotating shallow water equations with physical viscosity. We obtain the local existence of classical solutions without assuming the initial height is small or a small perturbation of some constant status. Moreover, the initial vacuum is allowed and the spatial measure of the set of vacuum can be arbitrarily large. In particular, the initial height can even have compact support; in this case, a blow-up example is given.

EXISTENCE OF SOLUTIONS OF QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS IN BANACH SPACES

We prove the local existence of classical solutions of quasi- linear integrodifierential equations in Banach spaces. The results are obtained by using fractional powers of operators and the Schauder flxed- point theorem. An example is provided to illustrate the theory.

GLOBAL EXPONENTIAL STABILITY OF CLASSICAL SOLUTIONS TO THE HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the high- and low-frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood–Paley decomposition and Bony's para-product formula.