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Local existence of smooth solutions for the semigeostrophic equations on curved domains
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Abstract
We prove local-in-time existence of smooth solutions to the semigeostrophic equations in the general setting of smooth, bounded and simply connected domains of endowed with an arbitrary conformally flat metric and non-vanishing Coriolis term. We present a construction taking place in Eulerian coordinates, avoiding the classical reformulation in dual variables, used in the flat case with constant Coriolis force, but lacking in this general framework.
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The aim of this note is to show that Alexandrov solutions of the Monge-Ampere
equation, with right hand side bounded away from zero and infinity, converge
strongly in if their right hand side converge strongly in
. As a corollary we deduce strong stability of
optimal transport maps.
The semigeostrophic equations are used in meteorology. They appear as a variant of the two-dimensional Euler incompressible equations in vorticity form, where the Poisson equation that relates the stream function and the vorticity field is just replaced by the fully nonlinear elliptic Monge-Ampère equation. This work gathers new results concerning the semigeostrophic equations: existence and stability of measure-valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, and convergence to the incompressible Euler equations.
In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampère equations, recently established by De Philippis and Figalli [1414.
De Philippis , G. ,
Figalli , A. W2, 1 regularity for solutions of the Monge-Ampère equation. Preprint. View all references], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.
This book counteracts the current fashion for theories of “chaos” and unpredictability by describing a theory that underpins the surprising accuracy of current deterministic weather forecasts, and it suggests that further improvements are possible. The book does this by making a unique link between an exciting new branch of mathematics called “optimal transportation” and existing classical theories of the large-scale atmosphere and ocean circulation. It is then possible to solve a set of simple equations proposed many years ago by Hoskins which are asymptotically valid on large scales, and use them to derive quantitative predictions about many large-scale atmospheric and oceanic phenomena. A particular feature is that the simple equations used have highly predictable solutions, thus suggesting that the limits of deterministic predictability of the weather may not yet have been reached. It is also possible to make rigorous statements about the large-scale behaviour of the atmosphere and ocean by proving results using these simple equations and applying them to the real system allowing for the errors in the approximation. There are a number of other titles in this field, but they do not treat this large-scale regime.
This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. Although the contents center on mathematical theory, many parts of the book showcase the interactions among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. The first half forms an introductory graduate course on vorticity and incompressible flow. The second half comprises a modern applied mathematics graduate course on the weak solution theory for incompressible flow. Andrew J. Majda is the Samuel Morse Professor of Arts and Sciences at the Courant Institute of Mathematical Sciences of New York University. He is a member of the National Academy of Sciences and has received numerous honors and awards includ- ing the National Academy of Science Prize in Applied Mathematics, the John von Neumann Prize of the American Mathematical Society and an honorary Ph.D. degree from Purdue University. Majda is well known for both his theoretical contributions to partial differential equations and his applied contributions to diverse areas besides in- compressible flow such as scattering theory, shock waves, combustion, vortex motion and turbulent diffusion. His current applied research interests are centered around Atmosphere/Ocean science. Andrea L. Bertozzi is Professor of Mathematics and Physics at Duke University. She has received several honors including a Sloan Research Fellowship (1995) and the Presidential Early Career Award for Scientists and Engineers (PECASE). Her research accomplishments in addition to incompressible flow include both theoretical and applied contributions to the understanding of thin liquid films and moving contact lines.
Hoskins's semigcostrophic equations are reformulated as a coupled Monge-Ampère/ transport problem [B. J. Hoskins, Quart. J. Royal Met. Soc., 97 (1971), pp. 139-153]. Existence of global weak solutions is obtained for this formulation.
Consideration of flows in which the rate of change of momentum is much
smaller than the Coriolis force suggests that the advected quantity
momentum may be approximated by its geostrophic value but that
trajectories cannot be so approximated. The resulting set of equations
imply full forms of the equations for potential temperature,
three-dimensional vorticity, potential vorticity and energy. A
transformation of horizontal coordinates products the `semi-geostrophic'
system in which conservation of potential vorticity and potential
temperature suffice to determine the motion. The system is capable of
describing the formation of fronts, jets, and the growth of baroclinic
waves into the nonlinear regime. It sheds some light on the success and
failure of the quasi-geostrophic equations.
Pure and applied mathematics series 140, repr
- Robert A Adams
Robert A. Adams. Sobolev spaces. eng. 2nd ed., [Repr.] Pure and applied mathematics series 140,
repr. 2008. Amsterdam [etc: Academic Press, 2008. isbn: 9780120441433.
Lectures on elliptic partial differential equations. eng. Appunti Lecture Notes 18
- Luigi Ambrosio
Luigi Ambrosio. Lectures on elliptic partial differential equations. eng. Appunti Lecture Notes 18.
Pisa: Edizioni della Normale, 2018. isbn: 9788876426506.