The formation of singularities in relativistic flows is not well understood.Smooth solutions to the relativistic Euler equations are known to have a finite lifespan;the possible breakdown mechanisms are shock formatio...The formation of singularities in relativistic flows is not well understood.Smooth solutions to the relativistic Euler equations are known to have a finite lifespan;the possible breakdown mechanisms are shock formation,violation of the subluminal conditions andmass concentration.We propose a new hybrid Glimm/centralupwind scheme for relativistic flows.The scheme is used to numerically investigate,for a family of problems,which of the above mechanisms is involved.展开更多
The formation of singularity and breakdown of classical solutions to the three- dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite...The formation of singularity and breakdown of classical solutions to the three- dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite-time formation of singularity in classical solu- tions is proved for certain initial data. For the compressible viscoelastic fluids, a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.展开更多
In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,...In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,μ(ρ)=ρ^(δ),δ>1).We first obtain the local existence of the regular solution,then show that the regular solution will blow up in finite time if initial data have an isolated mass group,no matter how small and smooth the initial data are.It is worth mentioning that based on the transport structure of some intrinsic variables,we obtain the L^(∞)bound of the density,which helps to remove the restrictionδ≤γin Li-Pan-Zhu[21]and Huang-Wang-Zhu[13].展开更多
We are concerned with singularity formation of strong solutions to the two-dimensional(2D)full compressible magnetohydrodynamic equations with zero resistivity in a bounded domain.By energy method and critical Sobolev...We are concerned with singularity formation of strong solutions to the two-dimensional(2D)full compressible magnetohydrodynamic equations with zero resistivity in a bounded domain.By energy method and critical Sobolev inequalities of logarithmic type,we show that the strong solution exists globally if the temporal integral of the maximum norm of the deformation tensor is bounded.Our result is the same as Ponce’s criterion for 3D incompressible Euler equations.In particular,it is independent of the magnetic field and temperature.Additionally,the initial vacuum states are allowed.展开更多
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
The author gets a blow-up result of C1 solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.
In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity （〉 1）, a blow-up re...In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity （〉 1）, a blow-up result is obtained for the C^1 solution to the Cauchy problem under the assumptions where there is a simple genuinely nonlinear characteristic and the initial data possess certain weaker decaying properties.展开更多
For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the for...For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.展开更多
In this paper,we consider the Cauchy problem for pressureless gases in two space dimensions with the generic smooth initial data(density and velocity).These equations give rise to singular curves,where the mass has a ...In this paper,we consider the Cauchy problem for pressureless gases in two space dimensions with the generic smooth initial data(density and velocity).These equations give rise to singular curves,where the mass has a positive density with respect to the 1-dimensional Hausdorff measure.We observe that the system of equations describing these singular curves is not hyperbolic.For analytic data,local solutions are constructed by using a version of the Cauchy-Kovalevskaya theorem.We then study the interaction of two singular curves in the generic position.Finally,for a generic initial velocity field,we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed.展开更多
In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight...In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.展开更多
In this paper,the curve shortening flow in a general Riemannian manifold is studied,Altschuler’s results about the flow for space curves are generalized.For any n-dimensional(n ≥ 2)Riemannian manifold(M,g) with some...In this paper,the curve shortening flow in a general Riemannian manifold is studied,Altschuler’s results about the flow for space curves are generalized.For any n-dimensional(n ≥ 2)Riemannian manifold(M,g) with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.展开更多
基金supported by the National Science Foundation(NSF)through grants DMS-0811150.Y.Sun’s research is partially supported by NSF through the NSF Joint Institutes’Postdoctoral Fellowship at SAMSI and a USC startup fund.
文摘The formation of singularities in relativistic flows is not well understood.Smooth solutions to the relativistic Euler equations are known to have a finite lifespan;the possible breakdown mechanisms are shock formation,violation of the subluminal conditions andmass concentration.We propose a new hybrid Glimm/centralupwind scheme for relativistic flows.The scheme is used to numerically investigate,for a family of problems,which of the above mechanisms is involved.
基金supported in part by the National Science Foundationthe Office of Naval Research
文摘The formation of singularity and breakdown of classical solutions to the three- dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite-time formation of singularity in classical solu- tions is proved for certain initial data. For the compressible viscoelastic fluids, a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.
基金supported by the National Natural Science Foundation of China(12371221,12161141004,11831011)the Fundamental Research Funds for the Central Universities and Shanghai Frontiers Science Center of Modern Analysis.
文摘In this paper,we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum inℝ,where the viscosity depends on the density in a super-linear power law(i.e.,μ(ρ)=ρ^(δ),δ>1).We first obtain the local existence of the regular solution,then show that the regular solution will blow up in finite time if initial data have an isolated mass group,no matter how small and smooth the initial data are.It is worth mentioning that based on the transport structure of some intrinsic variables,we obtain the L^(∞)bound of the density,which helps to remove the restrictionδ≤γin Li-Pan-Zhu[21]and Huang-Wang-Zhu[13].
基金partially supported by National Natural Science Foundation of China(Nos.11901474,12371227)。
文摘We are concerned with singularity formation of strong solutions to the two-dimensional(2D)full compressible magnetohydrodynamic equations with zero resistivity in a bounded domain.By energy method and critical Sobolev inequalities of logarithmic type,we show that the strong solution exists globally if the temporal integral of the maximum norm of the deformation tensor is bounded.Our result is the same as Ponce’s criterion for 3D incompressible Euler equations.In particular,it is independent of the magnetic field and temperature.Additionally,the initial vacuum states are allowed.
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The author gets a blow-up result of C1 solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.
文摘In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity （〉 1）, a blow-up result is obtained for the C^1 solution to the Cauchy problem under the assumptions where there is a simple genuinely nonlinear characteristic and the initial data possess certain weaker decaying properties.
基金Supported by the National Natural Science Foundation of China(Grant No.10926162)the Fundamental Research Funds for the Central Universities(Grant No.2009B01314)the Natural Science Foundation of HohaiUniversity(Grant No.2009428011)
文摘For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.
基金supported by U.S.National Science Foundation(Grant No.DMS2006884)(singularities and error bounds for hyperbolic equations)supported by U.S.National Science Foundation(Grant Nos.DMS-2008504 and DMS-2306258)supported by Zhejiang Normal University(Grant Nos.YS304222929 and ZZ323205020522016004)。
文摘In this paper,we consider the Cauchy problem for pressureless gases in two space dimensions with the generic smooth initial data(density and velocity).These equations give rise to singular curves,where the mass has a positive density with respect to the 1-dimensional Hausdorff measure.We observe that the system of equations describing these singular curves is not hyperbolic.For analytic data,local solutions are constructed by using a version of the Cauchy-Kovalevskaya theorem.We then study the interaction of two singular curves in the generic position.Finally,for a generic initial velocity field,we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed.
文摘In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.
基金Supported by NSFC (Grant No. 11721101)National Key Research and Development Project SQ2020YFA070080。
文摘In this paper,the curve shortening flow in a general Riemannian manifold is studied,Altschuler’s results about the flow for space curves are generalized.For any n-dimensional(n ≥ 2)Riemannian manifold(M,g) with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.
