Well-posedness In Energy Space Research Articles

Well-posedness results for a generalized Klein-Gordon-Schrödinger system

We consider the Klein-Gordon-Schrödinger system i∂tψ + Δψ = ϕ2ψ − ϕψ, (□ + 1)ϕ = −2|ψ|2ϕ + |ψ|2 with additional cubic terms and Cauchy data ψ(0)=ψ0∈Hs(Rn),ϕ(0)=ϕ0∈Hk(Rn),and (∂tϕ)(0)=ϕ1∈Hk−1(Rn) in space dimensions n = 2 and n = 3. We prove the local existence, uniqueness, and continuous dependence on the data in Bourgain-Klainerman-Machedon spaces for low regularity data, e.g., for s=−18 and k=38+ϵ in the case n = 2 and s = 0 and for k=12+ϵ in the case n = 3. Global well-posedness in energy space is also obtained as a special case. Moreover, we show the “unconditional” uniqueness in the space ψ ∈ C0([0, T], Hs), ϕ∈C0([0,T],Hs+12)∩C1([0,T],Hs−12), if s>322 for n = 2 and s>12 for n = 3.

Global well-posedness in energy space of small amplitude solutions for klein-gordon-zakharov equation in three space dimension

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt−Δu+u=−nu, (x,t)∈ℝ3×ℝ+,ntt−Δu=Δ|u|2, (x,t)∈ℝ3×ℝ+,u(x,0)=u0(x), ∂tu(x,0)=u1(x), n(x,0)=n0(x), ∂tn(x,0)=n1(x),is considered. It is shown that it is globally well-posed in energy space H1×L2×L2×H−1 if small initial data (u0(x),u1(x),n0(x),n1(x))∈(H1×L2×L2×H−1). It answers an open problem: Is it globally well-posed in energy space H1×L2×L2×H−1 for 3D Klein-Gordon-Zakharov equation with small initial data [1,2]? The method in this article combines the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequalities). We mainly extend the spaces Fs and Ns in one dimension [3] to higher dimension.

Global well-posedness in energy space for the Chern–Simons–Higgs system in temporal gauge

The Cauchy problem for the Chern–Simons–Higgs system in the [Formula: see text]-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an [Formula: see text]-estimate for solutions of the wave equation, and also takes advantage of a null condition.

Wellposedness in energy space for the nonlinear Klein–Gordon–Schrödinger system

This paper is concerned with the wellposedness of the nonlinear Klein–Gordon–Schrödinger (NKGS) equations under multi-interactions in 3 dimensions. By using the vanishing viscosity techniques and the compactness arguments, we establish the existence of the global finite energy solutions for the NKGS equations. In addition, by introducing a time piecewise function with integral form, we prove uniqueness and continuous dependence on the initial data.

Well-posedness in energy space for the periodic modified Benjamin–Ono equation

We prove that the periodic modified Benjamin–Ono equation is locally well-posed in the energy space H1/2. This ensures the global well-posedness in the defocusing case. The proof is based on an Xs,b analysis of the system after a gauge transform.

Well-posedness in the energy space for non-linear system of wave equations with critical growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure $$ \left\{ \begin{gathered} u_{tt} - \Delta u = - F_1 (|u|^2 ,|v|^2 )u, \hfill \\ v_{tt} - \Delta v = - F_2 (|u|^2 ,|v|^2 )v, \hfill \\ \end{gathered} \right. $$ where there exists a function F(λ, µ) such that $$ \frac{{\partial F(\lambda ,\mu )}} {{\partial \lambda }} = F_1 (\lambda ,\mu ),\frac{{\partial F(\lambda ,\mu )}} {{\partial \mu }} = F_2 (\lambda ,\mu ). $$ By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in “Shatah and Struwe’s paper, Ann. of Math. 138, 503–518 (1993)”.

Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions

Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions