arXiv subjects

Steve Schochet

Publications and source records attributed to Steve Schochet.

5 recordsLinked to original sources

arXiv2026

Singular Limits of the Shallow Water Equations on the Sphere

Solutions of the slightly compressible shallow water equations on a rapidly rotating sphere are shown to be bounded uniformly when ratio of the Froude number to the Rossby number is bounded. Moreover, in the singular limit in which the ratio of those parameters remains fixed while they both tend to zero, solutions with well-prepared initial data tend to corresponding solutions of limit equations. A convergence result is also obtained for the three-scale singular limit in which the Froude number tends to zero faster than the Rossby number.

math.AP↗

arXiv2022

Compartmental limit of discrete Bass models on networks

We introduce a new method for proving the convergence and the rate of convergence of discrete Bass models on various networks to their respective compartmental Bass models, as the population size $M$ becomes infinite. In this method, the full set of master equations is reduced to a smaller system of equations, which is closed and exact. The reduced finite system is embedded into an infinite system, and the convergence of that system to the infinite limit system is proved using standard ODE estimates. Finally, an ansatz provides an exact closure of the infinite limit system, which reduces that system to the compartmental model. Using this method, we show that when the network is complete and homogeneous, the discrete Bass model converges to the original 1969 compartmental Bass model, at the rate of $1/M$. When the network is circular, however, the compartmental limit is different, and the rate of convergence is exponential in $M$. In the case of a heterogeneous network that consists of $K$ homogeneous groups, the limit is given by a heterogeneous compartmental Bass model, and the rate of convergence is $1/M$. Using this compartmental model, we show that when the heterogeneity in the external and internal influence parameters among the $K$ groups is positively monotonically related, heterogeneity slows down the diffusion.

math.CA↗

arXiv2020

Convergence Rate Estimates for the Low Mach and Alfvén Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfvén numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors' previous paper, is much more complicated than for the classical two-scale singular limits.

math.AP↗

arXiv2019

Shallow water equations on a fast rotating surface

We prove that for rotating shallow water equations on a surface of revolution with variable Coriolis parameter and vanishing Rossby and Froude numbers, the classical solution satisfies uniform estimates on a fixed time interval with no dependence on the small parameters. Upon a transformation using the solution operator associated with the large operator, the solution converges strongly to a limit for which the governing equation is given. We also characterize the kernel of the large operator and define a projection onto that kernel. With these tools, we are able to show that the time-averages of the solution are close to longitude-independent zonal flows and height field.

math.AP↗

arXiv2019

Three-Scale Singular Limits of Evolutionary PDEs

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero.

math.AP↗