In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmo...In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function q corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extreme points of q on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of q corresponds to a family of steady multiple vortex patches approaching it.展开更多
In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional （n ≥ 2） Euler equations for steady irrotational flow that may contain stagnation points...In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional （n ≥ 2） Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy Hloc^-1（Ω） compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional （n ≥ 2） Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension （n ≥ 3）.展开更多
Based on the Boussinesq assumption, derived are couple equations of free surface elevation and horizontal velocities for horizontal irrotational flow, and analytical expressions of the corresponding pressure and verti...Based on the Boussinesq assumption, derived are couple equations of free surface elevation and horizontal velocities for horizontal irrotational flow, and analytical expressions of the corresponding pressure and vertical velocity. After the free surface elevation and horizontal velocity at a certain depth are obtained by numerical method, the pressure and vertical velocity distributions can be obtained by simple calculation. The dispersion at different depths is the same at the O (epsilon) approximation. The wave amplitude will decrease with increasing time due to viscosity, but it will increase due to the matching of viscosity and the bed slope, thus, flow is unstable. Numerical or analytical results show that the wave amplitude, velocity and length will increase as the current increases along the wave direction. but the amplitude will increase, and the wave velocity and length will decrease as the water depth decreases.展开更多
基金supported by National Natural Science Foundation of China (Grant No.11331010)supported by National Natural Science Foundation of China (Grant No.11771469)Chinese Academy of Sciences (Grant No.QYZDJ-SSW-SYS021)。
文摘In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function q corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extreme points of q on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of q corresponds to a family of steady multiple vortex patches approaching it.
基金supported in part by NSFC (10825102) for distinguished youth scholarNational Basic Research Program of China (973 Program) under Grant No.2011CB808002
文摘In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional （n ≥ 2） Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy Hloc^-1（Ω） compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional （n ≥ 2） Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension （n ≥ 3）.
基金National Natural Science Foundation of China.(Grant No.19572077)
文摘Based on the Boussinesq assumption, derived are couple equations of free surface elevation and horizontal velocities for horizontal irrotational flow, and analytical expressions of the corresponding pressure and vertical velocity. After the free surface elevation and horizontal velocity at a certain depth are obtained by numerical method, the pressure and vertical velocity distributions can be obtained by simple calculation. The dispersion at different depths is the same at the O (epsilon) approximation. The wave amplitude will decrease with increasing time due to viscosity, but it will increase due to the matching of viscosity and the bed slope, thus, flow is unstable. Numerical or analytical results show that the wave amplitude, velocity and length will increase as the current increases along the wave direction. but the amplitude will increase, and the wave velocity and length will decrease as the water depth decreases.
