Global Existence Of Small Solutions Research Articles

Global existence of small solutions to the Kerr–Debye model for the three-dimensional Cauchy problem

Global existence of small solutions to the Kerr–Debye model for the three-dimensional Cauchy problem

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $u_{tt} + 2u_t - a_{ij} (u_t ,\nabla u)\partial _i \partial _j u = f$ corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation $ - a_{ij} (0,\nabla v)\partial _i \partial _j v = h$ . We then give conditions for the convergence, as t → ∞, of the solution of the evolution equation to its stationary state.

Global small amplitude solutions for two-dimensional nonlinear Klein–Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein–Gordon equations with masses m 1 , m 2 satisfying the resonance relation m 2 = 2 m 1 > 0 . We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O ( | t | − 1 ) as t → ± ∞ in L ∞ . In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288–323].

The global existence of small solutions to the Oldroyd-B model

The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫0T* ‖▿u(t)‖L∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.

The Cauchy problem for the elliptic–hyperbolic Davey–Stewartson system in Sobolev space

We study the initial value problem for the elliptic–hyperbolic Davey–Stewartson systems (0.1) { i ∂ t u + Δ u = c 1 | u | 2 u + c 2 u ∂ x 1 φ , ( t , x ) ∈ R 3 , ( ∂ x 1 2 − ∂ x 2 2 ) φ = ∂ x 1 | u | 2 , u ( 0 , x ) = ϕ ( x ) , where Δ = ∂ x 1 2 + ∂ x 2 2 , c 1 , c 2 ∈ R , u is a complex valued function and φ is a real valued function. Our purpose is to prove the local existence and uniqueness of the solution for (0.1) in the Sobolev space H 3 / 2 + ( R 2 ) with small mass. Our methods rely heavily on Hayashi and Hirata (1996) [11], but we improve partial results of it, which got global existence of small solutions to (0.1) in weighted Sobolev space H 3 , 0 ∩ H 0 , 3 . Our main new tools are Kenig–Ponce–Vega type commutator estimate in Kenig, Ponce and Vega (1993) [16] and its variant form.

Small solutions to semi-linear wave equations with radial data of critical regularity

This paper investigates the problem of global existence of small solutions to semi-linear wave equations with radially symmetric data of critical regularity. Under radial symmetry we focus on the case where the power of nonlinear term is somewhat smaller than the conformal power. Our result covers the case where the power is strictly larger than the John-Glassey exponent in two or three space dimensions. In higher dimension it applies to the equation whose power is strictly larger than the $L^2$-critical exponent. The main theorem is therefore an improvement over a previous result due to Lindblad and Sogge. The new ingredient in our proof is an effective use of some weighted estimates of radially symmetric solutions to inhomogeneous wave equations.

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in $${\mathbb{R}}^2$$

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.

Global Solutions for 3D Quadratic Schrodinger Equations

This is the firstof the three papers where we present a new method based on the concept of space-time resonance to prove global existence of small solutions to nonlinear dispersive equations. The idea is that time resonances (dynamical systems resonances) correspond to interactions between plane waves; but since for dispersive equations we deal with localized solutions, it is crucial to take also into account the traveling speeds of the different wave packets. Here we show how this idea, and the analytical method that this naturally suggests, leads to a simple proof of global existence and scattering for quadratic nonlinear Schrodinger equations in three dimensions.

Global existence and asymptotic behavior of solutions to systems of semilinear wave equations in two space dimensions

We consider the Cauchy problem for systems of semilinear wave equations in 2D with small initial data, and introduce a sufficient condition for global existence of small solutions. Our condition is weaker than the null condition for 2D wave equations, and it is motivated by Alinhac's condition for 3D. We also show that some global solutions under our condition are not asymptotically free.

Global Small Solutions of the Cauchy Problem for Nonisotropic Schrödinger Equations

Abstract. In this paper we study the existence of global small solutions of theCauchy problem for the non-isotropically perturbed nonlinear Schr¨odinger equation:iu t + ∆u + |u| α u + aP di u x i x x x = 0, where a is real constant, 1 ≤ d < n is a inte-ger, α is a positive constant, and x = (x 1 ,x 2 ,··· ,x n ) ∈ R n . For some admissible α weshow the existence of global(almost global) solutions and we calculate the regularity ofthose solutions. 1. IntroductionThis paper is concerned with the Cauchy problem of the following fourth-ordernonlinear dispersive equation in R n ×R:(1)ˆiu t +∆u+|u| α u+aP di u x i i i i = 0, x ∈ R n , t ∈ R.u(x,0) = ϕ(x), x ∈ R n .where d is an integer, 1 ≤ d < n, a is a nonzero real constants, and α is a pos-itive constant. This equation is a modified version of the semi-discrete nonlinearSchrodinger equation (see [1]), or a non-isotropic higher-order perturbation of thesecond-order nonlinear Schrodinger equation:(2) iu t +∆u+|u| α u = 0, x ∈ R n , t ∈ R.Clearly, the following equations are special cases of (1)iu

Nonlinear Schroedinger Equations with Radially Symmetric Data of Critical Regularity

This paper is concerned with the global existence of small solutions to pure-power nonlinear Schrodinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schrodinger equation has global in time solutions provided that the initial data have the small norm. The key ingredient in the proof of this result is an effective use of global weighted smoothing estimates specific to radially symmetric solutions.

The Global Existence of Small Solutions to the Incompressible Viscoelastic Fluid System in 2 and 3 Space Dimensions

In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this system blows up at some finite time T*, then it is necessary that Furthermore, we prove the global existence of smooth solutions to this system provided that initial data is sufficiently close to the equilibrium state.

A Remark on the Almost Global Existence Theorems of Keel, Smith and Sogge

We give a new proof of temporally global existence of small solutions for systems of semi-linear wave equations. Our proof uses the Klainerman-Sideris inequality and a space-time L2-estimate. We also discuss whether the scale-invariant version of the space-time L2-estimates can hold, and obtain some related estimates. Among other things, we prove that the Keel-Smith-Sogge estimate actually holds in all space dimensions.

GLOBAL EXISTENCE FOR A CLASS OF SYSTEMS OF NONLINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

Consider a system of nonlinear wave equations for i=1,⋯,m, where Fi (i=1,⋯,m) are smooth functions of degree 2 near the origin of their arguments, and u=(u1,⋯,um), while ∂u and ∂x ∂u represent the first and second derivatives of u, respectively. In this paper, the author presents a new class of nonlinearity for which the global existence of small solutions is ensured. For example, global existence of small solutions for will be established, provided that .

Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions

Consider a quasi-linear system of two Klein–Gordon equations with masses m 1, m 2. We prove that when m 1≠2 m 2 and m 2≠2 m 1, such a system has global solutions for small, smooth, compactly supported Cauchy data. This extends a result proved by Sunagawa (J. Differential Equations 192 (2) (2003) 308) in the semi-linear case. Moreover, we show that global existence holds true also when m 1=2 m 2 and a convenient null condition is satisfied by the nonlinearities.

On the asymptotics for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation where We prove the global existence of small solutions , if the initial data u 1 belong to some analytic function space and are sufficiently small. For the coefficients λ j we assume that there exists θ 0 > 0 such that for all and also we suppose that the initial data are such that where ϵ is a small positive constant depending on the size of initial function in a suitable norm. We also find the large time asymptotic formulas for solutions. In the short range region the solution has an additional logarithmic time decay comparing with the corresponding linear case.

Small global solutions for nonlinear Dirac equations

The global Cauchy problem for nonlinear Dirac and Klein-Gordon equations in space--time $\mathbb R^{n+1}$ is studied in Sobolev and Besov spaces. Global existence of small solutions is proved under a scale-invariant setting when reduced to the corresponding massless case.

Global Existence and Decay for Solutions to the “Bad” Boussinesq Equation in Two Space Dimensions

We study the global existence of small solutions for the “bad” Boussinesq equation with potential V(x) verifying We show that the supremum norm of the small solution of this equation decays to zero as t tends to infinity and when the degree of the nonlinear term possesses an appropriate lower bound.

Résultats d'existence pour les écoulements réguliers de fluides viscoélastiques incompressibles à loi différentielle de type White–Metzner en dimension 3

This paper is concerned with incompressible viscoelastic fluids which obey a differential constitutive law of White–Metzner type. We establish the existence and uniqueness of local solutions in 3-D as well as the global existence of small solutions. We then deduce the existence and asymptotic stability of small periodic and stationary solutions. Finally, we prove that the 2-D results obtained in Hakim (J. Math. Anal. Appl. 185 (1994) 675–705) remain true without any restriction on the smallness of the retardation parameter which is the linking coefficient between the equation of velocity (Navier–Stokes equation) and the transport equation verified by the extra-stress tensor. To cite this article: L. Molinet, R. Talhouk, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Time-Space Estimates of Solutions to General Semilinear Parabolic Equations

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$. We give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations. Moreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.