This paper resolves a longstanding discussion of a mathematical problem important in contaminant hydrogeology and chemical-reaction engineering, by discussing the foundations for a conceptual model of a dilute miscible solute undergoing longitudinal convection and dispersion with moderate rates of appearance and disappearance in a finite continuum. It is demonstrated that: (i) Hulburts conditions (a first-type entrance with a third-type exit) fail to satisfy overall mass conservation; (ii) the conditions of Wehner and Wilhelm which reduce to those of Danckwerts (a third-type entrance with a zero-gradient exit) satisfy overall mass conservation yet fail to satisfy internal consistency with the governing equation; (iii) only third-type boundaries simultaneously satisfy internal consistency and overall mass conservation which are, respectively, a necessary and sufficient condition for any solution to the governing equation. This result is extensible to quite general governing equations since the boundary conditions are shown to be independent of the fate mechanisms.
We study two-dimensional Rayleigh-Bénard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number $\rm{Ra}$, this estimate interpolates between the Whitehead-Doering bound by $\rm{Ra}^{\frac{5}{12}}$ for free-slip conditions [13] and the classical Doering-Constantin $\rm{Ra}^{\frac{1}{2}}$ bound [4].
We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we g…
A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature…
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } b \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $. In particular, compared with the…
We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a …
In this paper we study the hydrodynamic limit of a B.G.K. like kinetic model on domains with boundaries via $BV_{loc}$ theory. We obtain as a consequence existence results for scalar multidimensional conservation laws with kinetic boundary conditions. We require that the initial and boundary data satisfy the optimal assumptions that they all belong to $L^1\cap L^\infty$ with the additional regularity assumptions that the initial data are in $BV_{loc}$. We also extend our hydrodynamic analysis to the case of a generalized kinetic model to account for forces effects and we obtain as a consequenc…
The initial value problem for the conservation law $\partial_t u+(-Δ)^{α/2}u+\nabla \cdot f(u)=0$ is studied for $α\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(\sqrtε)$ in the $L^\in…
In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial co…
We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection-diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model [G. Allaire et al, Chemical Engineering Science, 65 (2010), pp.2292-2300], …
We prove an existence theorem for the Boltzmann-Fermi-Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.
