This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data i...This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.展开更多
A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole...A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.展开更多
In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible break...In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L^∞ (0, T; L^6)-norm of the density of particles and the ^L1(0, T; L^∞)-norm of the deformation tensor of velocity gradient.展开更多
In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global exis...In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions :and a general decay for the energy are established using the multiplier technique. Also, 'we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.展开更多
The gradient blowup of the equation ut = △u + a(x)|△u|p + h(x), where p 〉 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate ...The gradient blowup of the equation ut = △u + a(x)|△u|p + h(x), where p 〉 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.展开更多
This paper proves a new regularity criterion ω: = rotu∈L1(0,T;B·0∞,∞) for the 3D generalized MHD system with fractional diffusion terms(-Δ)αu with α >9/8 and zero magnetic diffusivity. Here u is the flu...This paper proves a new regularity criterion ω: = rotu∈L1(0,T;B·0∞,∞) for the 3D generalized MHD system with fractional diffusion terms(-Δ)αu with α >9/8 and zero magnetic diffusivity. Here u is the fluid velocity,ω is the vorticity and B·0∞,∞is the homogeneous Besov space.展开更多
In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al...In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.展开更多
In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the...In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.展开更多
We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical ...We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat.We study the influence of the advection on the blowup properties of the solutions and con-clude that large advection is not favorable for blowup.Moreover,we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.展开更多
Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in...Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in X.In this paper we show that the absolute orbifold Gromov-Witten theory ofXacan be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X,S and Da,the natural restriction homomorphism HCR^*(X)→HCR*(S)and the first Chern class of the tautological line bundle over DQ.To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of(Xa|Da)and(Na|Da).As applications of these results,we prove an orbifold version of a conjecture of Maulik and Pandharipande(Topology,2006)on the Gromov-Witten theory of blowups along complete intersections,a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You(J Pure Appl Algebra,2016).展开更多
In this article,we discuss the blowup phenomenon of solutions to the wdiffusion equation with Dirichlet boundary conditions on the graph.Through Banach fixed point theorem,comparison principle,construction of auxiliar...In this article,we discuss the blowup phenomenon of solutions to the wdiffusion equation with Dirichlet boundary conditions on the graph.Through Banach fixed point theorem,comparison principle,construction of auxiliary function and other methods,we prove the local existence of solutions,and under appropriate conditions the blowup time and blowup rate estimation are given.Finally,numerical experiments are given to illustrate the blowup behavior of the solution.展开更多
This article is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained.The...This article is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained.The asymptotic behavior of the free boundary problem is studied.Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong,while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.展开更多
For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H r...For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.展开更多
This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established...This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established. The uniqueness and nonuniqueness results for the system with vanishing initial value are given.展开更多
文摘This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.
文摘A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.
基金supported by the National Basic Research Program of China(973 Program)(2011CB808002)the National Natural Science Foundation of China(11371152,11128102,11071086,and 11571117)+3 种基金the Natural Science Foundation of Guangdong Province(S2012010010408)the Foundation for Distinguished Young Talents in Higher Education of Guangdong(2015KQNCX095)the Major Foundation of Hanshan Normal University(LZ201403)the Scientific Research Foundation of Graduate School of South China Normal University(2014ssxm04)
文摘In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L^∞ (0, T; L^6)-norm of the density of particles and the ^L1(0, T; L^∞)-norm of the deformation tensor of velocity gradient.
文摘In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions :and a general decay for the energy are established using the multiplier technique. Also, 'we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.
基金supported by Youth Foundation of NSFC (10701061)Fundamental Research Funds for the Central Universities of ChinaScientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry
文摘The gradient blowup of the equation ut = △u + a(x)|△u|p + h(x), where p 〉 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.
文摘This paper proves a new regularity criterion ω: = rotu∈L1(0,T;B·0∞,∞) for the 3D generalized MHD system with fractional diffusion terms(-Δ)αu with α >9/8 and zero magnetic diffusivity. Here u is the fluid velocity,ω is the vorticity and B·0∞,∞is the homogeneous Besov space.
基金supported by National Natural Science Foundation of China(Grant Nos.12231016 and 12071391)Guangdong Basic and Applied Basic Research Foundation(Grant No.2022A1515010860)。
文摘In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.
基金The research of Gaowei Cao was supported in part by the NSFC(Grant 11701551 and Grant 11971024)the China Scholarship Council No.202004910200.The research of Hui Kan was supported in part by the NSFC(Grant 11801551)+1 种基金The research of Wei Xiang was supported in part by the Research Grants Council of the HKSAR,China(Project No.City U 11332916,Project No.City U 11304817 and Project No.City U 11303518)The research of X.Z.Yang was supported in part by the NSFC(Grant 11471332)。
文摘In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.
基金supported by Natural Science Foundation of China(No.11901238)Natural Science Foundation of Shandong Province(No.ZR2019MA063).
文摘We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat.We study the influence of the advection on the blowup properties of the solutions and con-clude that large advection is not favorable for blowup.Moreover,we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.
基金supported by National Natural Science Foundation of China(Grant Nos.11890663,11821001,11826102 and 11501393)the Sichuan Science and Technology Program(Grant No.2019YJ0509)a joint research project of Laurent Mathematics Research Center of Sichuan Normal University and V.C.&V.R.Key Lab of Sichuan Province。
文摘Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in X.In this paper we show that the absolute orbifold Gromov-Witten theory ofXacan be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X,S and Da,the natural restriction homomorphism HCR^*(X)→HCR*(S)and the first Chern class of the tautological line bundle over DQ.To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of(Xa|Da)and(Na|Da).As applications of these results,we prove an orbifold version of a conjecture of Maulik and Pandharipande(Topology,2006)on the Gromov-Witten theory of blowups along complete intersections,a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You(J Pure Appl Algebra,2016).
文摘In this article,we discuss the blowup phenomenon of solutions to the wdiffusion equation with Dirichlet boundary conditions on the graph.Through Banach fixed point theorem,comparison principle,construction of auxiliary function and other methods,we prove the local existence of solutions,and under appropriate conditions the blowup time and blowup rate estimation are given.Finally,numerical experiments are given to illustrate the blowup behavior of the solution.
基金supported by NRF of Korea (Grant No.2009-0094070)POSTECH BK21 and National Natural Science Foundation of China (Grant Nos.10671172,10801115)
文摘This article is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained.The asymptotic behavior of the free boundary problem is studied.Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong,while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.
基金Project supported by the Zheng Ge Ru FoundationTianyuan Foundation of China
文摘For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10171088)and also by SRF for ROCS,SEM.
文摘This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established. The uniqueness and nonuniqueness results for the system with vanishing initial value are given.