Fractional Diffusion Limit for Collisional Kinetic Equations

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.     

A Local and Global Well-Posedness Results for the General Stress-Assisted Diffusion Systems

We prove the local and global in time existence of the classical solutions to two general classes of the stress-assisted diffusion systems. Our results are applicable in the context of the non-Euclidean elasticity and liquid crystal elastomers.     

Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L ₁ model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic L ^p theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.     

Local Well-Posedness and Blowup Criterion of the Boussinesq Equations in Critical Besov Spaces

We consider the existence of the 2D inviscid Boussinesq equations in critical Besov spaces and obtain some blowup criteria.     

Well-Posedness of the Hydrostatic MHD Equations

The well-posedness of the equations of fluid mechanics in the hydrostatic limit is well known to be a difficult problem. Partial results, both positive and negative, will be reviewed below. In this paper, it is shown that, for ideal magnetohydrodynamics, a magnetic field parallel to the flow direction can ensure well-posedness. The only condition required is that the flow is subalfvenic. The result has some relevance to viscoelastic flows of the upper convected Maxwell fluid, which, in the infinite Weissenberg number limit, is related to ideal MHD.     

Local Well-Posedness for Fluid Interface Problems

In this paper, we prove the local well-posedness of the fluid interface problem with surface tension where the velocity fields are not assumed to be irrotational and the fluid domains are not assumed to be simply connected. Viewed as a Lagrangian system with the configuration space being an infinite dimensional manifold possessing many symmetries, this problem is reduced to the evolution of the interface, determined by its mean curvature, and the evolution of the rotational part of the velocity fields, determined by the symmetries. This framework also applies to several other fluid surface problems which are outlined in the paper.     

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

Archive for Rational Mechanics and Analysis - We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ to the nonlinear fractional diffusion equation, $${\partial_t u...     

Local and Global Well-Posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity

In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with H ² initial data.     

Boundary Stabilization and Disturbance Rejection for Time Fractional Order Diffusion–Wave Equations

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.     

Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations ${(-\Delta )^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}}$ with an isolated singularity, where ${\sigma\in (0,1)}$ . We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved by Caffarelli et al. (Comm Pure Appl Math 42:271–297, 1989).     

Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data \({(u_0,\theta_0)}\) are required to be only in the space \({X=\{f\in L^2(\mathbb{R}^2)\,|\,\partial_{x} f \in L^2(\mathbb{R}^2)\}}\), and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in \({H^2(\mathbb{R}^2)}\). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the \({L^\infty(\mathbb{R}^2)}\) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.     

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.     

Well-Posedness of Boundary Layer Equations for Time-Dependent Flow of Non-Newtonian Fluids

We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument.     

Application of Fractional Differential Equations for Modeling the Anomalous Diffusion of Contaminant from Fracture into Porous Rock Matrix with Bordering Alteration Zone

Solute diffusion from a fracture into a porous rock with an altered zone bordering the fracture is modeled by a system of two diffusion equations (one for the altered zone and another for the intact porous matrix) with different coefficients of effective diffusivity. Since experimental studies of diffusion into rock samples with altered zones indicate that mathematical models of diffusion based on Fick’s law do not adequately describe the concentration field in a sample, fractional order diffusion equations are chosen in this study for modeling the anomalous mass transport in the rocks. In the case of significantly higher porosity of the altered zone (e.g., this is typical for carbonates) the effective diffusivity here can be much higher than the effective diffusivity of non-altered rocks. By introducing a small parameter that is the ratio of effective diffusivities in the non-altered and altered regions and applying the technique of perturbations, approximate analytical solutions for concentrations in the altered zone bordering the fracture and in the intact surrounding rocks are obtained. Based on these solutions, different regimes of diffusion into the rocks with different physical properties are modeled and analyzed. It is shown that, using experimentally obtained data, the orders of the fractional derivatives in the differential equations can be readily calibrated for the every specific rock.     

Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

Journal of Dynamics and Differential Equations - In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time...     

分数因子与分数阶完整力学系统的运动方程和循环积分

引入分数因子和分数增量,给出了分数阶微积分的定义和性质;基于分数阶导数的定义,证明了含有分数因子的等时变分与分数阶算子的交换关系;提出了分数阶完整保守和非保守系统的Hamilton原理;建立了分数阶完整保守系统和非保守系统的运动微分方程;得到了分数阶完整保守系统的循环积分;并利用分数阶循环积分导出分数阶罗兹方程．最后给出了两个例子．研究表明利用分数因子给出的分数阶微分方程是一个含有分数因子的通常的微分方程,那么分数阶系统运动微分方程的求解都可以采用通常微分方程的求解方法．     

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) =  C _γ ρ ^γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.     

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, \({\triangle^{\alpha/2}}\) for \({\alpha \in (0,2)}\) . Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on α in the BV-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits \({\alpha \downarrow 0}\) and \({\alpha \uparrow 2}\) . In the limit \({\alpha \uparrow 2}\) , \({\triangle^{\alpha/2}}\) converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231–251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).