Combining TT* argument and bilinear interpolation,this paper obtains the Strichartz and smoothing estimates of dispersive semigroup e^(-itP(D)) in weighted L^(2) spaces.Among other things,we recover the results in[1]....Combining TT* argument and bilinear interpolation,this paper obtains the Strichartz and smoothing estimates of dispersive semigroup e^(-itP(D)) in weighted L^(2) spaces.Among other things,we recover the results in[1].Moreover,the application of these results to the well-posedness of some equations are shown in the last section.展开更多
The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-LittlewoodSobolev ...The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-LittlewoodSobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMOx(Rn) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V(t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in Lr(Rn) is proved.展开更多
In the present paper, the full range Strichartz estimates for homogeneous Schroedinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz esti...In the present paper, the full range Strichartz estimates for homogeneous Schroedinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.展开更多
It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive e...It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive estimates cannot hold.This paper aims at establishing local dispersive estimates on H^(d) for the linear Schrodinger equation,by a refined study of the Schrodinger ker-nel St on H^(d).The sharpness of these estimates is discussed through several examples.Our approach,based on the explicit formula of the heat kernel on H^(d) derived by Gaveau[19],is achieved by combining complex analysis and Fourier-Heisenberg tools.As a by-product of our results we establish local Stri-chartz estimates and prove that the kernel St concentrates on quantized hori-zontal hyperplanes of H^(d).展开更多
In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as ...In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.展开更多
基金supported by the NSFC(12071437)the National Key R&D Program of China(2022YFA1005700).
文摘Combining TT* argument and bilinear interpolation,this paper obtains the Strichartz and smoothing estimates of dispersive semigroup e^(-itP(D)) in weighted L^(2) spaces.Among other things,we recover the results in[1].Moreover,the application of these results to the well-posedness of some equations are shown in the last section.
基金supported by National Natural Science Foundation of China(Grant Nos.11371057,11261051 and 11161042)Doctoral Fund of Ministry of Education of China(Grant No.20130003110003)the Fundamental Research Funds for the Central Universities(Grant No.2012CXQT09)
文摘The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-LittlewoodSobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMOx(Rn) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V(t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in Lr(Rn) is proved.
基金the Graduate Student Innovation Fund of Fudan University.
文摘In the present paper, the full range Strichartz estimates for homogeneous Schroedinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.
文摘It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive estimates cannot hold.This paper aims at establishing local dispersive estimates on H^(d) for the linear Schrodinger equation,by a refined study of the Schrodinger ker-nel St on H^(d).The sharpness of these estimates is discussed through several examples.Our approach,based on the explicit formula of the heat kernel on H^(d) derived by Gaveau[19],is achieved by combining complex analysis and Fourier-Heisenberg tools.As a by-product of our results we establish local Stri-chartz estimates and prove that the kernel St concentrates on quantized hori-zontal hyperplanes of H^(d).
基金supported by the NSFC(12331007)the National Key Research and Development Program of China(2020YFA0713803)。
文摘In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.
