ISSN:
1937-5093
eISSN:
1937-5077
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Kinetic & Related Models
March 2016 , Volume 9 , Issue 1
Issue on fluid dynamics and electromagnetism: Theory and numerical approximation
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2016, 9(1): 1-49
doi: 10.3934/krm.2016.9.1
+[Abstract](1936)
+[PDF](680.1KB)
Abstract:
We prove a quantitative propagation of chaos and entropic chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard [9] and Fournier [10] where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
We prove a quantitative propagation of chaos and entropic chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard [9] and Fournier [10] where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
2016, 9(1): 51-74
doi: 10.3934/krm.2016.9.51
+[Abstract](1995)
+[PDF](693.0KB)
Abstract:
The kinetic theory of fluid turbulence modeling developed by Degond and Lemou in [7] is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multi-dimensional prototype model and first propose an appropriate change of frame that makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a one-dimensional framework on the basis of the developed strategy to confirm its efficiency.
The kinetic theory of fluid turbulence modeling developed by Degond and Lemou in [7] is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multi-dimensional prototype model and first propose an appropriate change of frame that makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a one-dimensional framework on the basis of the developed strategy to confirm its efficiency.
2016, 9(1): 75-103
doi: 10.3934/krm.2016.9.75
+[Abstract](2620)
+[PDF](504.6KB)
Abstract:
In this paper, the compressible Navier-Stokes system (CNS) with constant viscosity coefficients is considered in three space dimensions. we prove the global existence of spherically symmetric weak solutions to the free boundary problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.
In this paper, the compressible Navier-Stokes system (CNS) with constant viscosity coefficients is considered in three space dimensions. we prove the global existence of spherically symmetric weak solutions to the free boundary problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.
2016, 9(1): 105-129
doi: 10.3934/krm.2016.9.105
+[Abstract](2013)
+[PDF](439.9KB)
Abstract:
In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium distributions is considered. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria.
In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium distributions is considered. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria.
2016, 9(1): 131-164
doi: 10.3934/krm.2016.9.131
+[Abstract](2134)
+[PDF](2196.3KB)
Abstract:
The present paper deals with the modeling of formation and destruction of swarms using a nonlinear Boltzmann--like equation. We introduce a new model that contains parameters characterizing the attractiveness or repulsiveness of individuals. The model can represent both gregarious and solitarious behaviors. In the latter case we provide a mathematical analysis in the space homogeneous case. Moreover we identify relevant hydrodynamic limits on a formal way. We introduce some preliminary results in the case of gregarious behavior and we indicate open problems for further research. Finally, we provide numerical simulations to illustrate the ability of the model to represent formation or destruction of swarms.
The present paper deals with the modeling of formation and destruction of swarms using a nonlinear Boltzmann--like equation. We introduce a new model that contains parameters characterizing the attractiveness or repulsiveness of individuals. The model can represent both gregarious and solitarious behaviors. In the latter case we provide a mathematical analysis in the space homogeneous case. Moreover we identify relevant hydrodynamic limits on a formal way. We introduce some preliminary results in the case of gregarious behavior and we indicate open problems for further research. Finally, we provide numerical simulations to illustrate the ability of the model to represent formation or destruction of swarms.
2016, 9(1): 165-191
doi: 10.3934/krm.2016.9.165
+[Abstract](2092)
+[PDF](504.0KB)
Abstract:
The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.
The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.
2016, 9(1): 193-215
doi: 10.3934/krm.2016.9.193
+[Abstract](2184)
+[PDF](904.9KB)
Abstract:
We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method. For time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme has the expected order up to the numerical noise from the optimization routine, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter.
We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method. For time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme has the expected order up to the numerical noise from the optimization routine, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter.
2016, 9(1): 217-235
doi: 10.3934/krm.2016.9.217
+[Abstract](2015)
+[PDF](413.7KB)
Abstract:
A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.
A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.
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