This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density ...This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density and initial energy, we prove that if ρ0 ∈ Lγ ∩ H3 with γ ∈ (1, 6), then the problem possesses a unique global classical solution on R^(3) × [0, T] with any T ∈ (0, ∞). It’s worth noting that both the vacuum states and possible random largeness of initial energy are allowed.展开更多
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate...We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.展开更多
In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. W...In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. We show that the life span of any smooth solution will not be extended to ∞, if the initial vacuum only appears in some local domain and the magnetic field vanishes on the interface that separates the vacuum and non-vacuum state, regardless the size of the initial data or the far field state.展开更多
In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying...This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C^1 solution to Cauchy problem.展开更多
By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result o...By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.展开更多
This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 i...This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 is obtained, and a sharp estimate of the life span for the classical solution is given.展开更多
This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do...This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do not depend on the radius r.Second,the classical solutions of the linearized model in R^(3) are obtained by combining the continuation and compactness methods.Finally,the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.展开更多
The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for ...The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.展开更多
In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ ...In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.展开更多
In this paper, we prove the existence of global classical solutions to time-dependent Ginzburg-Landau（TDGL） equations. By the properties of Besov and Sobolev spaces, together with the energy method, we establish the...In this paper, we prove the existence of global classical solutions to time-dependent Ginzburg-Landau（TDGL） equations. By the properties of Besov and Sobolev spaces, together with the energy method, we establish the global existence and uniqueness of classical solutions to the initial boundary value problem for time-dependent Ginzburg-Landau equations.展开更多
We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data ...We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.展开更多
The paper concerns Cauchy,problem for one-dimensional hydromagnetic dynamics with dissipative terms. When the dissipation coefficient is equal to zero it is shown that the smooth solutions develop shocks in the finite...The paper concerns Cauchy,problem for one-dimensional hydromagnetic dynamics with dissipative terms. When the dissipation coefficient is equal to zero it is shown that the smooth solutions develop shocks in the finite time if the initial amounts of entropy and magnetic field are smaller than those of sound waves; when it is larger than zero, and the initial amounts of entropyI this dissipation coefficient and the magnetic field in each period are smaller than those of sound Waves, then the smooth solutions blow up in the finite time. Moreover, the life-span of the smooth solution is given.展开更多
Global classical solutions near Maxwellians are constructed for the Boltzmann equation in a periodic box with angular soft cutoff, that is, -3 〈 γ 〈 0. The construction of global solution is based on an energy meth...Global classical solutions near Maxwellians are constructed for the Boltzmann equation in a periodic box with angular soft cutoff, that is, -3 〈 γ 〈 0. The construction of global solution is based on an energy method used in [9].展开更多
In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on...In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with ＂small＂ initial data.展开更多
In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 bou...In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.展开更多
In this paper,we consider a kind of quasilinear hyperbolic systems with inhomogeneous terms satisfying dissipative condition or matching condition.For the Cauchy problem of this kind of systems,we prove that,if the in...In this paper,we consider a kind of quasilinear hyperbolic systems with inhomogeneous terms satisfying dissipative condition or matching condition.For the Cauchy problem of this kind of systems,we prove that,if the initial data is small and satisfies some decay condition,and the system is weakly linearly degenerate,then the Cauchy problem admits a unique global classical solution on t ≥ 0.展开更多
The paper considers the Cauchy problem with small initial values for semilinear wave equations with weighted nonlinear terms.Similar to Strauss exponent p0(n)which is the positive root of the quadratic equation 1+1/2(...The paper considers the Cauchy problem with small initial values for semilinear wave equations with weighted nonlinear terms.Similar to Strauss exponent p0(n)which is the positive root of the quadratic equation 1+1/2(n+1)p-1/2(n-1)p^(2)=0,we get smaller critical exponents p_(m)(n),p_(m)^(*)(n)and have global existence in time when p>p_(m)(n)or p>p_(m)^(*)(n).In addition,for the blow-up case,the introduction of the spacial weight shows the optimality of new upper and lower bound.展开更多
Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ...Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ . M∞展开更多
We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially.We first prove the local existen...We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially.We first prove the local existence and uniqueness of the strong solutions,where the initial compatibility condition proposed by Cho et al.(2004),Cho and Kim(2006)and Choe and Kim(2003)is removed in a suitable sense.Then the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition.Moreover,for the initial data satisfying some additional regularity and the compatibility condition,the strong solution is proved to be a classical one.展开更多
基金supported by the Natural Science Foundation of Shandong Province of China(ZR2024MA033ZR2021QA049).
文摘This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density and initial energy, we prove that if ρ0 ∈ Lγ ∩ H3 with γ ∈ (1, 6), then the problem possesses a unique global classical solution on R^(3) × [0, T] with any T ∈ (0, ∞). It’s worth noting that both the vacuum states and possible random largeness of initial energy are allowed.
基金supported by National Natural Science Foundation of China (11001090)the Fundamental Research Funds for the Central Universities(11QZR16)
文摘We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.
基金supported in part by National Natural Science Foundation of China(11231006)Natural Science Foundation of Shanghai(14ZR1423100)China Scholarship Council
文摘In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. We show that the life span of any smooth solution will not be extended to ∞, if the initial vacuum only appears in some local domain and the magnetic field vanishes on the interface that separates the vacuum and non-vacuum state, regardless the size of the initial data or the far field state.
文摘In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
文摘This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C^1 solution to Cauchy problem.
文摘By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.
基金Project supported by the NSF of China! (19971O62)the NSF of Fujian Province!(A97020) the NSF of Educational Committee of
文摘This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 is obtained, and a sharp estimate of the life span for the classical solution is given.
基金supported by the National Natural Science Foundation of China(12026253,12026244,11971357)the Natural Science Foundation of Guangdong Province(2018A030310008,2021A1515010303)+6 种基金Guangdong Key Laboratory for Functional Substances in Medicinal Edible Resources and Healthcare Products(2021B1212040015)NSF of Guangdong Provincial Department of Education(2019KTSCX097)Chaozhou Science and Technology plan project(2019ZC02)supported by the Key Project of National Natural Science Foundation of China(12131010)the National Natural Science Foundation of China(11771155,11571117,11871005)the Natural Science Foundation of Guangdong Province(2017A030313003,2019A1515011491,2021A1515010249)the Science and Technology Program of Guangzhou(2019050001)。
文摘This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do not depend on the radius r.Second,the classical solutions of the linearized model in R^(3) are obtained by combining the continuation and compactness methods.Finally,the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.
基金This work is supported by the Natural Science Foundation of China(10071048) Excellent Young Teachers Program by the MOE of China
文摘The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Science Foundation in Higher Education of Henan(18A110036)
文摘In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.
基金Supported by National Natural Science Foundation of China(11201415,11571159)Program for New Century Excellent Talents in Fujian Province University(JA14191)
文摘In this paper, we prove the existence of global classical solutions to time-dependent Ginzburg-Landau（TDGL） equations. By the properties of Besov and Sobolev spaces, together with the energy method, we establish the global existence and uniqueness of classical solutions to the initial boundary value problem for time-dependent Ginzburg-Landau equations.
基金supported by the National Natural ScienceFoundation of China(11871024)the Fundamental Research Program of Shanxi Province(202103021223182)。
文摘We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.
基金Project supported by the National Natural Science Foundation of China(No. 10571024)the Natural Science Foundation of Henan Province of China (No.200510078005)the Science Foundation of Educational Department of Henan Province of China (No.200511051700)
文摘The paper concerns Cauchy,problem for one-dimensional hydromagnetic dynamics with dissipative terms. When the dissipation coefficient is equal to zero it is shown that the smooth solutions develop shocks in the finite time if the initial amounts of entropy and magnetic field are smaller than those of sound waves; when it is larger than zero, and the initial amounts of entropyI this dissipation coefficient and the magnetic field in each period are smaller than those of sound Waves, then the smooth solutions blow up in the finite time. Moreover, the life-span of the smooth solution is given.
文摘Global classical solutions near Maxwellians are constructed for the Boltzmann equation in a periodic box with angular soft cutoff, that is, -3 〈 γ 〈 0. The construction of global solution is based on an energy method used in [9].
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Henan Provincial Natural Science Foundation of China(152300410226)
文摘In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with ＂small＂ initial data.
基金supported by the National NaturalScience Foundation of China(11971069 and 12126307)。
文摘In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.
基金Supported by National Science Foundation of China（10671124）
文摘In this paper,we consider a kind of quasilinear hyperbolic systems with inhomogeneous terms satisfying dissipative condition or matching condition.For the Cauchy problem of this kind of systems,we prove that,if the initial data is small and satisfies some decay condition,and the system is weakly linearly degenerate,then the Cauchy problem admits a unique global classical solution on t ≥ 0.
基金Supported by National Natural Science Foundation (Grant No.71572156)
文摘The paper considers the Cauchy problem with small initial values for semilinear wave equations with weighted nonlinear terms.Similar to Strauss exponent p0(n)which is the positive root of the quadratic equation 1+1/2(n+1)p-1/2(n-1)p^(2)=0,we get smaller critical exponents p_(m)(n),p_(m)^(*)(n)and have global existence in time when p>p_(m)(n)or p>p_(m)^(*)(n).In addition,for the blow-up case,the introduction of the spacial weight shows the optimality of new upper and lower bound.
基金Project supported by the National Natural Science Foundation of China (No.10225102) the 973 Project of the Ministry of Science and Technology of China and the Doctoral Programme Foundation of the Ministry of Education of China.
文摘Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ . M∞
基金supported by National Natural Science Foundation of China(Grant Nos.11688101,11731007 and 11671412)Youth Innovation Promotion Association,Chinese Academy of Sciences。
文摘We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially.We first prove the local existence and uniqueness of the strong solutions,where the initial compatibility condition proposed by Cho et al.(2004),Cho and Kim(2006)and Choe and Kim(2003)is removed in a suitable sense.Then the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition.Moreover,for the initial data satisfying some additional regularity and the compatibility condition,the strong solution is proved to be a classical one.
