Mikhail Feldman

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following second-order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential φ : Ω ⊂ R → R: (1.1) div (ρ(|Dφ|)Dφ) = 0, where the density function ρ(q) is

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monges original criterion for efficiency [19] was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from Rn to Rn [23]. A second proof of this existence result formed the subject of a recent monograph by Evans and Gangbo [7], who avoided Sudakovs measure decomposition results by using a partial differential equations approach. In the present manuscript, we give a third existence proof for optimal mappings, which has some advantages (and disadvantages) relative to existing approaches: it requires no continuity or separation of the mass distributions, yet our explicit construction yields more geometrical control than the abstract method of Sudakov. (Indeed, this control turns out to be essential for addressing a gap which has recently surfaced in Sudakovs approach to the problem in dimensions n > 3; see the remarks at the end of this section.) It is also shorter and more flexible than either, and can be adapted to handle transportation on Riemannian manifolds or around obstacles, as we plan to show in a subsequent work [13]. The problem considered here is the classical one: Problem 1 (Monge). Fix a norm d(x, y) = llx-yll on Rn, and two densities -non-negative Borel functions f+, fE Ll (Rn) satisfying the mass balance condition

Monge’s transport problem on a Riemannian manifold

Monges problem refers to the classical problem of optimally transporting mass: given Borel probability measures μ + ¬= μ - , find the measurepreserving map s: M → M between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold M - and assuming absolute continuity of μ + - an optimal map will be shown to exist. Aspects of its uniqueness are also established.

Potential theory for shock reflection by a large-angle wedge

When a plane shock hits a wedge head on, it experiences a reflection, and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of reflected shocks may occur, including regular and Mach reflection. However, most fundamental issues for shock reflection phenomena have not been understood, such as the transition among the different patterns of shock reflection; therefore, it is essential to establish a global existence and stability theory for shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and stability of solutions to shock reflection, especially for potential flow, which has widely been used in aerodynamics. The theoretical problems involve several challenging difficulties in the analysis of nonlinear partial differential equations including elliptic-hyperbolic mixed type, free-boundary problems, and corner singularity, especially when an elliptic degenerate curve meets a free boundary. Here we develop a potential theory to overcome these difficulties and to establish the global existence and stability of solutions to shock reflection by a large-angle wedge for potential flow. The techniques and ideas developed will be useful to other nonlinear problems involving similar difficulties.

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for a steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity (non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Fréchet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.

Regularity of solutions to regular shock reflection for potential flow

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete. However, most of the fundamental issues for shock reflection have not been understood, including the regularity and transition of different patterns of shock reflection configurations. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in C1,1 across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in a physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle (the transonic boundary from the elliptic to hyperbolic region) and at the point where the pseudo-sonic circle (the degenerate elliptic curve) meets the reflected shock (a free boundary connecting the elliptic to hyperbolic region). In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the C1,1-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the C2,α regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region. The techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties.

A Geometric Level-Set Formulation of a Plasma-Sheath Interface

In this paper, we present a new geometric level-set formulation of a plasma-sheath interface arising in plasma physics. We formally derive the explicit dynamics of the interface from the Euler-Poisson equations and study the local-time evolution of the interface and sheath in some special cases.

On Lagrangian Solutions for the Semi-geostrophic System with Singular Initial Data

We show that weak (Eulerian) solutions for the semi-geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in dual space. However, such solutions are physically relevant to the model. Thus, we discuss a natural generalization of weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We prove existence of such solutions in the case of discrete measures in dual space. We also prove that weak Lagrangian solutions in physical space determine solutions in the dual space. This implies conservation of geostrophic energy along the Lagrangian trajectories in the physical space.

Prandtl-Meyer reflection for supersonic flow past a solid ramp

We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu’s theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. We apply new mathematical techniques developed in our recent work to obtain monotonicity properties and uniform apriori estimates for weak solutions, which allow us to employ the LeraySchauder degree argument to complete the theory for the general case.

SELF-SIMILAR ISOTHERMAL IRROTATIONAL MOTION FOR THE EULER, EULER–POISSON SYSTEMS AND THE FORMATION OF THE PLASMA SHEATH

We consider the self-similar dynamics of a compressible irrotational isothermal fluid and show that a variant of Elling–Lius ellipticity principle holds. Specially we prove that a closed pseudo-Mach surface is inconsistent with smooth flow. Furthermore, we give an application of our result to the formulation of a plasma sheath in a plasma consisting of cold ions and hot electrons.