Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional int...Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).展开更多
We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-...We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.展开更多
In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner(GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions o...In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner(GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function f, we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.展开更多
In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^(s(·))p(·),h(·),q(R^(3))with s(·)=4-2...In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^(s(·))p(·),h(·),q(R^(3))with s(·)=4-2α-3/p(·).We prove global well-posedness result with small initial data belonging to FN^(4-2α-3/p(·))p(·),h(·)q(R^(3)).The result of this paper extends some recent work.展开更多
In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-...In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.展开更多
In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To...In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.展开更多
We investigate the solution of an N-unit series system with finite number of vacations. By using C0-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic soluti...We investigate the solution of an N-unit series system with finite number of vacations. By using C0-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.展开更多
In this paper we prove the local well-posedness of strong solutions to a chemotaxisshallow water system with initial vacuum in a bounded domainΩ■R^(2)without the standard compatibility condition for the initial data...In this paper we prove the local well-posedness of strong solutions to a chemotaxisshallow water system with initial vacuum in a bounded domainΩ■R^(2)without the standard compatibility condition for the initial data.This improves some results obtained in[J.Differential Equations 261(2016),6758-6789].展开更多
We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function s...We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.展开更多
We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in R^(d)(d=2,3).By exploiting the intrinsic structure of the equations and using harmonic analysis tools(especially the ...We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in R^(d)(d=2,3).By exploiting the intrinsic structure of the equations and using harmonic analysis tools(especially the Littlewood-Paley theory),we prove the global solutions to this system with small initial data restricted in the Sobolev spaces.Moreover,the initial temperature may vanish at infinity.展开更多
In this paper,we prove the global well-posedness of the 2 D Boussinesq equations with three kinds of partial dissipation;among these the initial data(u_(0),θ_(0))is required such that its own and the derivative of on...In this paper,we prove the global well-posedness of the 2 D Boussinesq equations with three kinds of partial dissipation;among these the initial data(u_(0),θ_(0))is required such that its own and the derivative of one of its directions(x,y)are assumed to be L^(2)(R^(2)).Our results only need the lower regularity of the initial data,which ensures the uniqueness of the solutions.展开更多
Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven mo...Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.展开更多
In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly prope...In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly properly efficient solutions of set-valued optimization problem.展开更多
We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and...We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and the existence of the unique positive dynamic solution of the system.展开更多
In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of in...In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of integro-differential equations of first kind originating from an aeroelasticity problem. By manipulating the bounds of initial conditions with random variations, this study numerically demonstrated the well-posedness properties of the equations. Finally, an assumption of separating variables, allowed for linear splines to be chosen as a basis and for the differentiation and integration of the integro-differential part to be interchanged;hence, a numerical scheme was constructed.展开更多
In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids wi...In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids with the initial(angular) velocity being located in sub-critical Sobolev spaces H^(s)(R^(2))(0<s<1) and the initial density being bounded from above and below by some positive constants. The global unique solvability of the 2D incompressible inhomogeneous asymmetric fluids with the initial data in the critical Besov space(u_(0), w_(0))∈˙B^(0)_(2,1)(R^(2))andρ^(−1)−1∈˙B^(ε)_(2/ε),1(R^(2))is established. In particular, the uniqueness of the solution is also obtained without any more regularity assumptions on the initial density which is an improvement on the recent result of Abidi and Gui(2021) for the 2D inhomogeneous incompressible NavierStokes system.展开更多
In this paper,we study the well-posedness and blow-up solutions for the fractional Schrodinger equation with a Hartree-type nonlinearity together with a power-type subcritical or criticai perturbations.For nonradial i...In this paper,we study the well-posedness and blow-up solutions for the fractional Schrodinger equation with a Hartree-type nonlinearity together with a power-type subcritical or criticai perturbations.For nonradial initial data or radial initial data,we prove the local well-posedness for the defocusing and the focusing cases with sub-critical or critical nonlinearity.We obtain the global well-posedness for the defocusing case,and for the focusing mass-subcritical case or mass-critical case with initial da-ta small enough.We also investigate blow-up solutions for the focusing mass-critical problem.展开更多
The well-posedness of the dynamic framework in earth-system model(ESM for short)is a common issue in earth sciences and mathematics.In this paper,the authors first introduce the research history and fundamental roles ...The well-posedness of the dynamic framework in earth-system model(ESM for short)is a common issue in earth sciences and mathematics.In this paper,the authors first introduce the research history and fundamental roles of the well-posedness of the dynamic framework in the ESM,emphasizing the three core components of ESM,i.e.,the atmospheric general circulation model(AGCM for short),land-surface model(LSM for short)and oceanic general circulation model(OGCM for short)and their couplings.Then,some research advances made by their own research group are outlined.Finally,future research prospects are discussed.展开更多
In this paper,we study the subcritical dissipative quasi-geostrophic equa-tion.By using the Littlewood Paley theory,Fourier analysis and standard techniques we prove that there exists a unique global-in-time solution ...In this paper,we study the subcritical dissipative quasi-geostrophic equa-tion.By using the Littlewood Paley theory,Fourier analysis and standard techniques we prove that there exists a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces FN^(3-2a+(λ-2)/p)_(p,λ,q).Moreover,we show the asymptotic behavior of the global solution v.i.e.||v(t)||FN^(3-2a+(λ-2)/p)_(p,λ,q)decays to zero as time goes to infinity.展开更多
基金the NSF of China(12171266,12171062)the NSF of Chongqing(CSTB2022NSCQ-JQX0004)。
文摘Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).
基金supported by the NationalNatural Science Foundation of China(12001236)the Natural Science Foundation of Guangdong Province(2020A1515110494)。
文摘We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.
基金Yuhui Chen was supported by the NNSF of China(12201655)Qinghe Yao was supported by the NNSF of China (11972384)+2 种基金the Guangdong Science and Technology Fund (2021B1515310001)Zheng-an Yao was supported by the NNSF of China (11971496)the National Key R&D Program of China (2020YFA0712500)。
文摘In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner(GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function f, we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.
文摘In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^(s(·))p(·),h(·),q(R^(3))with s(·)=4-2α-3/p(·).We prove global well-posedness result with small initial data belonging to FN^(4-2α-3/p(·))p(·),h(·)q(R^(3)).The result of this paper extends some recent work.
文摘In this paper, some theoretical notions of well-posedness and of well-posedness in the generalized sense for scalar optimization problems are presented and some important results are analysed. Similar notions of well-posedness, respectively for a vector optimization problem and for a variational inequality of differential type, are discussed subsequently and, among the various vector well-posedness notions known in the literature, the attention is focused on the concept of pointwise well-posedness. Moreover, after a review of well-posedness properties, the study is further extended to a scalarizing procedure that preserves well-posedness of the notions listed, namely to a result, obtained with a special scalarizing function, which links the notion of pontwise well-posedness to the well-posedness of a suitable scalar variational inequality of differential type.
文摘In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.
文摘We investigate the solution of an N-unit series system with finite number of vacations. By using C0-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.
基金supported by NSFC(11971234)supported in part by NSFC(11671193)A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘In this paper we prove the local well-posedness of strong solutions to a chemotaxisshallow water system with initial vacuum in a bounded domainΩ■R^(2)without the standard compatibility condition for the initial data.This improves some results obtained in[J.Differential Equations 261(2016),6758-6789].
基金W.-X.Li's research was supported by NSF of China(11871054,11961160716,12131017)the Natural Science Foundation of Hubei Province(2019CFA007)T.Yang's research was supported by the General Research Fund of Hong Kong CityU(11304419).
文摘We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.
基金supported by the National Natural Science Foundation of China(11801090 and 12161004)Jiangxi Provincial Natural Science Foundation,China(20212BAB211004)+3 种基金supported by the National Natural Science Foundation of China(12171493)supported by the National Natural Science Foundation of China(11601533)Guangdong Provincial Natural Science Foundation,China(2022A1515011977)the Science and Technology Program of Shenzhen under grant 20200806104726001.
文摘We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in R^(d)(d=2,3).By exploiting the intrinsic structure of the equations and using harmonic analysis tools(especially the Littlewood-Paley theory),we prove the global solutions to this system with small initial data restricted in the Sobolev spaces.Moreover,the initial temperature may vanish at infinity.
基金partially supported by key research grant of the Academy for Multidisciplinary Studies,CNUsupported by NSFC(11901040)+1 种基金Beijing Municipal Commission of Education(KM202011232020)Beijing Natural Science Foundation(1204030)。
文摘In this paper,we prove the global well-posedness of the 2 D Boussinesq equations with three kinds of partial dissipation;among these the initial data(u_(0),θ_(0))is required such that its own and the derivative of one of its directions(x,y)are assumed to be L^(2)(R^(2)).Our results only need the lower regularity of the initial data,which ensures the uniqueness of the solutions.
基金Project supported by the National Natural Science Foundation of China(No.11672131)。
文摘Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.
文摘In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly properly efficient solutions of set-valued optimization problem.
文摘We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and the existence of the unique positive dynamic solution of the system.
文摘In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of integro-differential equations of first kind originating from an aeroelasticity problem. By manipulating the bounds of initial conditions with random variations, this study numerically demonstrated the well-posedness properties of the equations. Finally, an assumption of separating variables, allowed for linear splines to be chosen as a basis and for the differentiation and integration of the integro-differential part to be interchanged;hence, a numerical scheme was constructed.
基金supported by Natural Science Foundation of Zhejiang Province (Grant No. LY20A010017), Natural Science Foundation of Zhejiang Province (Grant No. LDQ23A010001)supported by National Natural Science Foundation of China (Grant No. 11931010)。
文摘In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids with the initial(angular) velocity being located in sub-critical Sobolev spaces H^(s)(R^(2))(0<s<1) and the initial density being bounded from above and below by some positive constants. The global unique solvability of the 2D incompressible inhomogeneous asymmetric fluids with the initial data in the critical Besov space(u_(0), w_(0))∈˙B^(0)_(2,1)(R^(2))andρ^(−1)−1∈˙B^(ε)_(2/ε),1(R^(2))is established. In particular, the uniqueness of the solution is also obtained without any more regularity assumptions on the initial density which is an improvement on the recent result of Abidi and Gui(2021) for the 2D inhomogeneous incompressible NavierStokes system.
基金This research is supported by NSFC key project under the grant number 11831003NSFC under the grant numbers 11971356 and 11571118by Fundamental Research Founds for the Central Universities under the grant numbers 2019MS110 and 2019MS112.
文摘In this paper,we study the well-posedness and blow-up solutions for the fractional Schrodinger equation with a Hartree-type nonlinearity together with a power-type subcritical or criticai perturbations.For nonradial initial data or radial initial data,we prove the local well-posedness for the defocusing and the focusing cases with sub-critical or critical nonlinearity.We obtain the global well-posedness for the defocusing case,and for the focusing mass-subcritical case or mass-critical case with initial da-ta small enough.We also investigate blow-up solutions for the focusing mass-critical problem.
基金supported by the National Natural Science Foundation of China(Nos.41975129,41630530)Key Research Program of Frontier Sciences,Chinese Academy of Sciences(No.QYZDYSSW-DQC002)the National Key Scientific and Technological Infrastructure project“Earth System Science Numerical Simulator Facility”(EarthLab).
文摘The well-posedness of the dynamic framework in earth-system model(ESM for short)is a common issue in earth sciences and mathematics.In this paper,the authors first introduce the research history and fundamental roles of the well-posedness of the dynamic framework in the ESM,emphasizing the three core components of ESM,i.e.,the atmospheric general circulation model(AGCM for short),land-surface model(LSM for short)and oceanic general circulation model(OGCM for short)and their couplings.Then,some research advances made by their own research group are outlined.Finally,future research prospects are discussed.
文摘In this paper,we study the subcritical dissipative quasi-geostrophic equa-tion.By using the Littlewood Paley theory,Fourier analysis and standard techniques we prove that there exists a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces FN^(3-2a+(λ-2)/p)_(p,λ,q).Moreover,we show the asymptotic behavior of the global solution v.i.e.||v(t)||FN^(3-2a+(λ-2)/p)_(p,λ,q)decays to zero as time goes to infinity.
