The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on RN,and also presents Liouville type theorems for the incompressible and compressible fluid equations.
This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a sig...This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.展开更多
This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employ...This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employing different time and space scalings.The paper shows that,depending on the magnitude of the parameters which define the scaling,the macroscopic quantities(number density,mean velocity and local temperature)are solutions of the acoustic equation,the linear incompressible Euler equation and the incompressible Navier–Stokes equation.The derivation is formally tackled by the recent moment method proposed by[C.Bardos,et al.,J.Stat.Phys.63(1991)323]and the results generalize the analysis performed in[C.Bianca,et al.,Commun.Nonlinear Sci.Numer.Simulat.29(2015)240].展开更多
This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of...This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.展开更多
In this paper,we use the Wigner meaure approach to study the semiclassical limit of nonlinear Schroedinger equation in small time.We prove that:the linits of the quantum density:ρ^ε=:|ψ^ε|^2,and the quantum moment...In this paper,we use the Wigner meaure approach to study the semiclassical limit of nonlinear Schroedinger equation in small time.We prove that:the linits of the quantum density:ρ^ε=:|ψ^ε|^2,and the quantum momentum:J^ε=:εIm(-↑ψ^ε↓△ψ^ε）satisfy the compressible Euler equations before the formation of singularities in the limit system.展开更多
The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are ...The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are taken into account. A typical example of nonconvex con-stitutive equation for fluids is Van der Waals' equation. The first order terms of these partialdifferential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class ofnonconvex equations of state, an existence theorem of traveling waves solutions with arbitrarylarge amplitude is established here. The authors distinguish between classical (compressive) andnonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequali-ties, and are characterized by the so-called kinetic relation, whose properties are investigatedin this paper.展开更多
基金Porject supported by the National Natural Science Foundation of China(11371240,11771274)the Natural Science Foundation of Shandong Pronvince of China(ZR2015AL006)
基金Project supported by the National Natural Science Foundation of China(11371240,11771274)the Shanghai Municipal Education Commission of Scientific Research Innovation Project(11ZZ84)
基金Project supported by KRF Grant (MOEHRD,Basic Research Promotion Fund)
文摘The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on RN,and also presents Liouville type theorems for the incompressible and compressible fluid equations.
基金supported by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE(No.EP/E035027/1)the UK EPSRC Award to the EPSRC Centre for Doctoral Training in PDEs(No.EP/L015811/1)+1 种基金the National Natural Science Foundation of China(No.10728101)the Royal Society-Wolfson Research Merit Award(UK)
文摘This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.
文摘This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employing different time and space scalings.The paper shows that,depending on the magnitude of the parameters which define the scaling,the macroscopic quantities(number density,mean velocity and local temperature)are solutions of the acoustic equation,the linear incompressible Euler equation and the incompressible Navier–Stokes equation.The derivation is formally tackled by the recent moment method proposed by[C.Bardos,et al.,J.Stat.Phys.63(1991)323]and the results generalize the analysis performed in[C.Bianca,et al.,Commun.Nonlinear Sci.Numer.Simulat.29(2015)240].
基金supported by National Natural Science Foundation of China(Grant No.11271184)China Scholarship Council,the Priority Academic Program Development of Jiangsu Higher Education Institutions,the Tsz-Tza Foundation,and Ministry of Science and Technology(Grant No.104-2628-M-006-003-MY4)
文摘This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.
文摘In this paper,we use the Wigner meaure approach to study the semiclassical limit of nonlinear Schroedinger equation in small time.We prove that:the linits of the quantum density:ρ^ε=:|ψ^ε|^2,and the quantum momentum:J^ε=:εIm(-↑ψ^ε↓△ψ^ε）satisfy the compressible Euler equations before the formation of singularities in the limit system.
文摘The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are taken into account. A typical example of nonconvex con-stitutive equation for fluids is Van der Waals' equation. The first order terms of these partialdifferential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class ofnonconvex equations of state, an existence theorem of traveling waves solutions with arbitrarylarge amplitude is established here. The authors distinguish between classical (compressive) andnonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequali-ties, and are characterized by the so-called kinetic relation, whose properties are investigatedin this paper.
基金Supported by National Natural Science Foundation of China(11261035 and 11171038)Science Research Foundation ofInstitute of Higher Education of Inner Mongolia Autonomous Region,China(NJZZ12198)+1 种基金Nature Science Foundation of InnerMongolia Autonomous Region,China(2012MS0102)Science and Technology Development Foundation of CAEP(2013A0202011)
基金Supported by the National Natural Science Foundation of China(11761054,11261035,11571002)the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT-15-A07)+3 种基金the Natural Science Foundation of Inner Mongolia Autonomous Region,China(2015MS0108,2012MS0102)the Science Research Foundation of Institute of Higher Education of Inner Mongolia Autonomous Region,China(NJZZ12198,NJZZ16234,NJZZ16235)Science and Technology Development Foundation of CAEP(2015B0101021)Defense Industrial Technology Development Program(B1520133015)
基金Support by the Special Funds of State Major Basic Research Projects(Grant No.1999075107)Innovation Funds of AMSS,CAS of China+1 种基金Support by the Austrian government START-prize project"Nonlinear SchrSdingerQuantum Boltzmann Equations"(Y-137-TEC)
