The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 ×...The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 if small initial data (u0 (x), u1 (x), n0 (x), n1 (x)) ∈ (H^1 ×L^2× L^2 × H^-1). It answers an open problem: Is it globally well-posed in energy space H^1 × L^2 × L^2 × 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 F^s and N^3 in one dimension [3] to higher dimension.展开更多
In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);...In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);n≥2 It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in H^(s)(R^(n))(s>n/2)for any large initial data.Thus this result can compare with that in one dimension except for the endpoint space H^(n/2).展开更多
文摘The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space is considered. It is shown that it is globally well-posed in energy space H^1 × L^2 × L^2 × H^-1 if small initial data (u0 (x), u1 (x), n0 (x), n1 (x)) ∈ (H^1 ×L^2× L^2 × H^-1). It answers an open problem: Is it globally well-posed in energy space H^1 × L^2 × L^2 × 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 F^s and N^3 in one dimension [3] to higher dimension.
文摘In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);n≥2 It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in H^(s)(R^(n))(s>n/2)for any large initial data.Thus this result can compare with that in one dimension except for the endpoint space H^(n/2).