ISSN:
1937-5093
eISSN:
1937-5077
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Kinetic & Related Models
June 2014 , Volume 7 , Issue 2
Issue on workshop in fluid mechanics and population dynamics
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2014, 7(2): 205-218
doi: 10.3934/krm.2014.7.205
+[Abstract](2424)
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Abstract:
The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.
The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.
2014, 7(2): 219-251
doi: 10.3934/krm.2014.7.219
+[Abstract](2238)
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Abstract:
In this paper we revisit the derivation of boundary conditions for the Boltzmann Equation. The interaction between the wall atoms and the gas molecules within a thin surface layer is described by a kinetic equation introduced in [10] and used in [1]. This equation includes a Vlasov term and a linear molecule-phonon collision term and is coupled with the Boltzmann equation describing the evolution of the gas in the bulk flow. Boundary conditions are formally derived from this model by using classical tools of kinetic theory such as scaling and systematic asymptotic expansion. In a first step this method is applied to the simplified case of a flat wall. Then it is extented to walls with nanoscale roughness allowing to obtain more complex scattering patterns related to the morphology of the wall. It is proved that the obtained scattering kernels satisfy the classical imposed properties of non-negativeness, normalization and reciprocity introduced by Cercignani [13].
In this paper we revisit the derivation of boundary conditions for the Boltzmann Equation. The interaction between the wall atoms and the gas molecules within a thin surface layer is described by a kinetic equation introduced in [10] and used in [1]. This equation includes a Vlasov term and a linear molecule-phonon collision term and is coupled with the Boltzmann equation describing the evolution of the gas in the bulk flow. Boundary conditions are formally derived from this model by using classical tools of kinetic theory such as scaling and systematic asymptotic expansion. In a first step this method is applied to the simplified case of a flat wall. Then it is extented to walls with nanoscale roughness allowing to obtain more complex scattering patterns related to the morphology of the wall. It is proved that the obtained scattering kernels satisfy the classical imposed properties of non-negativeness, normalization and reciprocity introduced by Cercignani [13].
2014, 7(2): 253-290
doi: 10.3934/krm.2014.7.253
+[Abstract](2726)
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Abstract:
In this work we study the stochastic process of two-species coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called well--stirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays self-similar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.
In this work we study the stochastic process of two-species coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called well--stirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays self-similar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.
2014, 7(2): 291-304
doi: 10.3934/krm.2014.7.291
+[Abstract](3122)
+[PDF](407.9KB)
Abstract:
In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1,3,9,11,17]. Precisely, we first prove that if for any $ i,\,j,\,k\in \{1,2,3\}$ there holds $ (\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k}) \in L_T^{\alpha,\gamma}$ with $\frac{2}{\alpha}+\frac{3}{\gamma}\leq 1+\frac{1}{\gamma},~2\leq \gamma\leq \infty $, then the solution $(u,b)$ is smooth on $\mathbb{R}^3\times(0,T]$. Secondly, we show that any component (resp. components) of $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k})$ in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space $L_T^{\alpha',\gamma'}$with $\frac{2}{\alpha'}+\frac{3}{\gamma'}\leq 1$, $3< \gamma'\leq \infty$. Fianlly, we obtain a Ladyzhenskaya-Prodi-Serrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].
In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1,3,9,11,17]. Precisely, we first prove that if for any $ i,\,j,\,k\in \{1,2,3\}$ there holds $ (\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k}) \in L_T^{\alpha,\gamma}$ with $\frac{2}{\alpha}+\frac{3}{\gamma}\leq 1+\frac{1}{\gamma},~2\leq \gamma\leq \infty $, then the solution $(u,b)$ is smooth on $\mathbb{R}^3\times(0,T]$. Secondly, we show that any component (resp. components) of $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k})$ in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space $L_T^{\alpha',\gamma'}$with $\frac{2}{\alpha'}+\frac{3}{\gamma'}\leq 1$, $3< \gamma'\leq \infty$. Fianlly, we obtain a Ladyzhenskaya-Prodi-Serrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].
2014, 7(2): 305-339
doi: 10.3934/krm.2014.7.305
+[Abstract](2534)
+[PDF](529.8KB)
Abstract:
In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.
In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.
2014, 7(2): 341-360
doi: 10.3934/krm.2014.7.341
+[Abstract](2293)
+[PDF](476.4KB)
Abstract:
This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of $F_t$ (the $L^2$ norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that $F_t$ satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.
This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of $F_t$ (the $L^2$ norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that $F_t$ satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.
2014, 7(2): 361-379
doi: 10.3934/krm.2014.7.361
+[Abstract](2719)
+[PDF](256.1KB)
Abstract:
The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offspring. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offspring. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
2014, 7(2): 381-400
doi: 10.3934/krm.2014.7.381
+[Abstract](2471)
+[PDF](413.5KB)
Abstract:
A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field limit of the particle system is obtained. Furthermore, the time-asymptotic flocking behavior of the solution to the kinetic equation is also derived. The technics used for local wellposedness and stability follow from similar ideas to those have been used in [3,14,22]. While in order to extend the local result globally, the main contribution here is to generate a series of new estimates for this Vlasov type equation, which imply that the growing of the characteristics can be controlled globally. Further estimates also show the long time flocking phenomena.
A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field limit of the particle system is obtained. Furthermore, the time-asymptotic flocking behavior of the solution to the kinetic equation is also derived. The technics used for local wellposedness and stability follow from similar ideas to those have been used in [3,14,22]. While in order to extend the local result globally, the main contribution here is to generate a series of new estimates for this Vlasov type equation, which imply that the growing of the characteristics can be controlled globally. Further estimates also show the long time flocking phenomena.
2014, 7(2): 401-414
doi: 10.3934/krm.2014.7.401
+[Abstract](2588)
+[PDF](391.0KB)
Abstract:
We show that the residual measures the difference in $L^1$ between the solutions to two different Boltzmann models of rarefied gases. This work is an extension of earlier work by Ha on the stability of Boltzmann's model, and more specifically on a nonlinear interaction functional that controls the growth of waves. The two kinetic models that are compared in this research are given by (possibly different) inverse power laws, such as the hard spheres and pseudo-Maxwell models. The main point of the estimate is that the modeling error is measured a posteriori, that is to say, the difference between the solutions to the first and second model can be bounded by a term that depends on only one of the two solutions. This work allows the stability estimate to be used to assess uncertainty, modeling or numerical, present in the solution of the first model without solving the second model.
We show that the residual measures the difference in $L^1$ between the solutions to two different Boltzmann models of rarefied gases. This work is an extension of earlier work by Ha on the stability of Boltzmann's model, and more specifically on a nonlinear interaction functional that controls the growth of waves. The two kinetic models that are compared in this research are given by (possibly different) inverse power laws, such as the hard spheres and pseudo-Maxwell models. The main point of the estimate is that the modeling error is measured a posteriori, that is to say, the difference between the solutions to the first and second model can be bounded by a term that depends on only one of the two solutions. This work allows the stability estimate to be used to assess uncertainty, modeling or numerical, present in the solution of the first model without solving the second model.
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