Small Data Solutions Research Articles

On the Benefit of Different Additional Regularity for the Weakly Coupled Systems of Semilinear Effectively Damped Waves

This paper is devoted to study the global existence of small data solutions to the Cauchy problem for weakly coupled systems of semilinear effectively damped waves, where the data have different additional regularities and different power nonlinearities.

On one-dimension semi-linear wave equations with null conditions

It is well-known that in space-dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in space-dimension two such result still holds for a certain class of quasi-linear wave equations with null conditions. The proofs are based on the decay mechanism of linear waves. However, in one space-dimension, waves do not decay. Nevertheless, we will prove that small data still lead to global solutions if the null condition is satisfied.

Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects

In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in $$\mathbb {R}^3$$ . In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions.

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.

Small data solutions of 2-D quasilinear wave equations under null conditions

For the 2-D quasilinear wave equation (∂t2-Δx)u+∑i,j=02gij(∂u)∂iju=0 satisfying null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation (∂t2-Δx)u+∑i,j=02gij(u, ∂u)∂iju=0 satisfying null conditions with small initial data and the coefficients depending simultaneously on u and ∂u. Through construction of an approximate solution, combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.

Regularity theory and global existence of small data solutions to semi-linear de Sitter models with power non-linearity

In this paper we consider the Cauchy problem for semi-linear de Sitter models with power non-linearity. The model of interest is ϕtt−e−2tΔϕ+nϕt+m2ϕ=|ϕ|p,(ϕ(0,x),ϕt(0,x))=(f(x),g(x)),where m2 is a non-negative constant. We study the global (in time) existence of small data solutions. In particular, we show the interplay between the power p, admissible data spaces and admissible spaces of solutions (in weak sense, in sense of energy solutions or in classical sense).

Nonlinear thermoelastic plate equations – Global existence and decay rates for the Cauchy problem

We consider the Cauchy problem in Rn for quasilinear thermoelastic Kirchhoff-type plate equations where the heat conduction is modeled by either the Cattaneo law or by the Fourier law. Additionally, we take into account possible inertial effects. Considering nonlinearities which are of fourth-order in the space variable, we deal with a quasilinear system which triggers difficulties typical for nonlinear Schrödinger equations. The different models considered are systems of mixed type comparable to Schrödinger–parabolic or Schrödinger–hyperbolic systems. The main task consists in proving sophisticated a priori estimates leading to obtaining the global existence of solutions for small data, neither known nor expected for the Cauchy problem in pure plate theory nor available before for the coupled system under investigation, where only special cases (bounded domains within with analytic semigroup setting, or the Cauchy problem with semilinear nonlinearities) had been treated before.

Galilean invariance and entropy principle for a system of balance laws of mixture type

After defining, in analogy with a mixture of continuous media, a system of balance laws of mixture type , we study the general properties obtained by imposing the Galilean invariance principle. For constitutive equations of local type we study also the entropy principle and we prove the compatibility between the two principles. These general results permit us to construct, from a single constituent theory, the corresponding theory of mixtures in an easy way. As an illustrative example of the general theory, we write down the hyperbolic system of balance laws of mixtures in which each component has 6 fields (mass density, velocity, temperature and dynamic pressure, among which only the last one is a nonequilibrium variable). This is the simplest system after Eulerian mixtures. Global existence of smooth solutions for small initial data is also proved.

On semilinear Tricomi equations with critical exponents or in two space dimensions

This paper is a complement of our recent works on the semilinear Tricomi equations in [9] and [10]. For the semilinear Tricomi equation ∂t2u−tΔu=|u|p with initial data (u(0,⋅),∂tu(0,⋅))=(u0,u1), where t≥0, x∈Rn (n≥3), p>1, and ui∈C0∞(Rn) (i=0,1), we have shown in [9] and [10] that there exists a critical exponent pcrit(n)>1 such that the solution u, in general, blows up in finite time when 1<p<pcrit(n), and there is a global small solution for p>pcrit(n). In the present paper, firstly, we prove that the solution of ∂t2u−tΔu=|u|p will generally blow up for the critical exponent p=pcrit(n) and n≥2, secondly, we establish the global existence of small data solution to ∂t2u−tΔu=|u|p for p>pcrit(n) and n=2. Thus, we have given a systematic study on the blowup or global existence of small data solution u to the equation ∂t2u−tΔu=|u|p for n≥2.

Global solvability of the inhomogeneous Ericksen–Leslie system with only bounded density

The most established theory for modeling the dynamics of nematic liquid crystals is the celebrated Ericksen–Leslie system, which presents some major analytical challenges. We study a simplified version of the system, which still exhibits the major difficulties. We consider the density-dependent case and study the Cauchy problem in the whole space. We establish the global existence of solutions for small initial data by assuming only that the initial density is bounded and kept away far from vacuum, while the initial velocity and the gradient of the initial director field belong to certain critical Besov spaces. Under slightly more assumptions on the initial velocity and the director field, we also prove that the solutions are unique.

The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position–velocity potential holomorphic coordinates, and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.

Well-posedness and scattering for the mass-energy NLS on ℝ n × ℳ k

Abstract We study the nonlinear Schrödinger equation posed on product spaces ℝ n × ℳ k {{\mathbb{R}}^{n}\times{\mathcal{M}}^{k}} , for n ≥ 1 {n\geq 1} and k ≥ 1 {k\geq 1} , with ℳ k {{\mathcal{M}}^{k}} any k-dimensional compact Riemannian manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of k = 2 {k=2} , H 1 {H^{1}} -scattering is also obtained.

The critical exponent for the dissipative plate equation with power nonlinearity

In this paper, we find the critical exponent of global small data solutions for a damped plate equation with power nonlinearity utt−Δutt+Δ2u+ut=|u|p,t≥0,x∈R2,and for a system of two weakly coupled damped plate equations. We show how assuming small data in the energy space H2×H1 and in L1 is sufficient to compensate the regularity-loss type decay effect created by the rotational inertia term −Δutt.

On the global solution problem for semilinear generalized Tricomi equations, I

In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $\partial_t^2 u-t^m \Delta u=|u|^p$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot))= (u_0, u_1)$, where $t\geq 0$, $x\in{\mathbb R}^n$ ($n\ge 3$), $m\in\mathbb N$, $p>1$, and $u_i\in C_0^{\infty}({\mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{\text{crit}}(m,n)>1$ such that the solution $u$, in general, blows up in finite time when $1<p<p_{\text{crit}}(m,n)$. We further show that there exists a conformal exponent $p_{\text{conf}}(m,n)> p_{\text{crit}}(m,n)$ such that the solution $u$ exists globally when $p>p_{\text{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{\text{crit}}(m,n)<p\leq p_{\text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.

Small data global existence for a fluid-structure model

We address the system of partial differential equations modeling motion of an elastic body inside an incompressible fluid. The fluid is modeled by the incompressible Navier–Stokes equations while the structure is represented by the damped wave equation with interior damping. The additional boundary stabilization γ, considered in our previous paper, is no longer necessary. We prove the global existence and exponential decay of solutions for small initial data in a suitable Sobolev space.

Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type

We study a class of $3D$ quadratic Schrödinger equations as follows, $(\partial_t -i Δ) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah in [2], the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1× 1 \to 0$ type interaction (the size of input frequency is '1' and the size of output frequency is '0'). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1× 0\to 1$ type interaction. The issue of strong $1× 1\to 0$ type interaction makes the global existence problem very delicate.In this paper, we show that, as long as there are '$ε$' derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i Δ)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi [3]. Instead of using vector fields, we consider this problem purely in Fourier space.

Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data

We prove peeling estimates for the small data solutions of the Maxwell Klein Gordon equations with non-zero charge and with a non-compactly supported scalar field, in $(3+1)$ dimensions. We obtain the same decay rates as in an earlier work by Lindblad and Sterbenz, but giving a simpler proof. In particular we dispense with the fractional Morawetz estimates for the electromagnetic field, as well as certain space-time estimates. In the case that the scalar field is compactly supported we can avoid fractional Morawetz estimates for the scalar field as well. All of our estimates are carried out using the double null foliation and in a gauge invariant manner.

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension. First, the pointwise estimates of solutions are obtained, furthermore, we obtain the optimal Lp, 1 ≤ p ≤ + ∞, convergence rate of solutions for small initial data. Then we establish the local existence of solutions, the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data. Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velazquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordstrom system.

Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells

We consider a coupled bulk/surface model for advection and diffusion of interacting chemical species in biological cells. Specifically, we consider a signalling protein that can exist in both a cytosolic and a membrane-bound state, along with a variable that gives a coarse-grained description of the cytoskeleton. The main focus of our work is on the well-posedness of the model, whereby the coupling at the boundary is the main source of analytical difficulty. A prioriLp-estimates, together with classical Schauder theory, deliver global existence of classical solutions for small data on bounded, Lipschitz domains. For two physically reasonable regularised versions of the boundary coupling, we are able to prove global existence of solutions for arbitrary data. In addition, we prove the existence of a family of steady-state solutions of the main model which are parametrised by the total mass of the membrane-bound signal molecule.