In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dime...In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.展开更多
We show that the spatial L q-norm(q>5/3)of the vorticity of an incompressible viscous fluid in R^3 remains bounded uniformly in time,provided that the direction of vorticity is Hölder continuous in space,and t...We show that the spatial L q-norm(q>5/3)of the vorticity of an incompressible viscous fluid in R^3 remains bounded uniformly in time,provided that the direction of vorticity is Hölder continuous in space,and that the space-time L q-norm of vorticity is finite.The Hölder index depends only on q.This serves as a variant of the classical result by Constantin-Fefferman(Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,Indiana Univ.J.Math.42(1993),775-789),and the related work by Grujić-Ruzmaikina(Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE,Indiana Univ.J.Math.53(2004),1073-1080).展开更多
In this paper,we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equa...In this paper,we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations.First,it is shown that there exists a positive constantεsuch that the solution(ρ,u,θ)to the full compressible Navier-Stokes equations can be extended beyond t=T provided that one of the following two conditions holds:(1)ρ∈L^(∞)(0,T;L^(∞)(R^(3)),u∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||u||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3;(0.1)(2)λ<3μ,ρ∈L^(∞)(0,T;L^(∞)(R^(3)),θ∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||θ||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3/2(0.2)To the best of our knowledge,this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid.Second,we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T^(*):(1)assuming that the pair(p,q) satisfies 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and (1.17),then lim sup_((t→T*))||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||u||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞;(0.3)(2)letting the pair(p,q)satisfy 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and(1.17),then lim sup (t→T*)||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||θ||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞=∞,(λ<3μ).(0.4)Third,without the condition on p in(0.1)and(0.3),the results also hold for the 3 D nonhomogeneous incompressible Navier-Stokes equations.The appearance of a vacuum in these systems could be allowed.展开更多
We consider a Neumann problem driven by a(p(z), q(z))-Laplacian(anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti...We consider a Neumann problem driven by a(p(z), q(z))-Laplacian(anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on R_(+)=(0,+∞).展开更多
In present work studied the new boundary value problem for semi linear(Power-type nonlinearities)system equations of mixed hyperbolic-elliptic Keldysh type in the multivariate dimension with the changing time directio...In present work studied the new boundary value problem for semi linear(Power-type nonlinearities)system equations of mixed hyperbolic-elliptic Keldysh type in the multivariate dimension with the changing time direction.Considered problem and equation belongs to the modern level partial differential equations.Applying methods of functional analysis,topological methods,“ε”-regularizing.and continuation by the parameter at the same time with aid of a prior estimates,under assumptions conditions on coefficients of equations of system,the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev's space.In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations,moreover in this case constructed new weithged spaces,the identity between of strong and weak solutions is established.展开更多
基金Supported by the National Natural Science Foundation of China (No. 10971174)
文摘In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.
文摘We show that the spatial L q-norm(q>5/3)of the vorticity of an incompressible viscous fluid in R^3 remains bounded uniformly in time,provided that the direction of vorticity is Hölder continuous in space,and that the space-time L q-norm of vorticity is finite.The Hölder index depends only on q.This serves as a variant of the classical result by Constantin-Fefferman(Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,Indiana Univ.J.Math.42(1993),775-789),and the related work by Grujić-Ruzmaikina(Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE,Indiana Univ.J.Math.53(2004),1073-1080).
基金supported by the NationalNatural Science Foundation of China(11971446,12071113,11601423,11771352,11871057,11771423,11671378,11701145)Project funded by China Postdoctoral Science Foundation(2020M672196)。
文摘In this paper,we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations.First,it is shown that there exists a positive constantεsuch that the solution(ρ,u,θ)to the full compressible Navier-Stokes equations can be extended beyond t=T provided that one of the following two conditions holds:(1)ρ∈L^(∞)(0,T;L^(∞)(R^(3)),u∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||u||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3;(0.1)(2)λ<3μ,ρ∈L^(∞)(0,T;L^(∞)(R^(3)),θ∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||θ||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3/2(0.2)To the best of our knowledge,this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid.Second,we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T^(*):(1)assuming that the pair(p,q) satisfies 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and (1.17),then lim sup_((t→T*))||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||u||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞;(0.3)(2)letting the pair(p,q)satisfy 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and(1.17),then lim sup (t→T*)||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||θ||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞=∞,(λ<3μ).(0.4)Third,without the condition on p in(0.1)and(0.3),the results also hold for the 3 D nonhomogeneous incompressible Navier-Stokes equations.The appearance of a vacuum in these systems could be allowed.
基金supported by National Natural Science Foundation of China (Grant No.12071413)Natural Science Foundation of Guangxi Grant No.2023GXNSFAA026085the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement (No.823731) CONMECH。
文摘We consider a Neumann problem driven by a(p(z), q(z))-Laplacian(anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on R_(+)=(0,+∞).
文摘In present work studied the new boundary value problem for semi linear(Power-type nonlinearities)system equations of mixed hyperbolic-elliptic Keldysh type in the multivariate dimension with the changing time direction.Considered problem and equation belongs to the modern level partial differential equations.Applying methods of functional analysis,topological methods,“ε”-regularizing.and continuation by the parameter at the same time with aid of a prior estimates,under assumptions conditions on coefficients of equations of system,the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev's space.In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations,moreover in this case constructed new weithged spaces,the identity between of strong and weak solutions is established.