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 C ^1,1 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 C ^1,1-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 C ^2,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.     

Time-dependent Stokes and Navier–Stokes problems with boundary conditions involving pressure, existence and regularity

This paper is concerned with the time-dependent Stokes and Navier–Stokes problems with nonstandard boundary conditions: the pressure is given on some part of the boundary. The stationary case was first studied by Bégue, Conca, Murat and Pironneau and, next, their study were completed by Bernard, mainly about regularity. In this paper, the Stokes problem is studied by a method analogous to that of Temam for the standard problem, combined with regularity results of Bernard for the nonstandard stationary case. We obtain existence, uniqueness and regularity H². In addition, in two dimensions, a regularity W^2,r, r2, is proved. Next, for the nonstandard Navier–Stokes problem, we present some existence, uniqueness and regularity H² results. The proof of existence is based on a fixed point method.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

Multidimensional viscous shocks II: The small viscosity limit

In this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit. Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes x_N/? dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L² estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

Regularity of weak solutions of Maxwell's equations with mixed boundary‐conditions

In this paper global H^s‐ and L^p‐regularity results for the stationary and transient Maxwell equations with mixed boundary conditions in a bounded spatial domain are proved. First it is shown that certain elements belonging to the fractional‐order domain of the Maxwell operator belong to H^s(Ω) for sufficiently small s > 0. It follows from this regularity result that H^s(Ω) is an invariant subspace of the unitary group corresponding to the homogeneous Maxwell equations with mixed boundary conditions. In the case that a possibly non‐linear conductivity is present a L^p‐regularity theorem for the transient equations is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

Finding a best traveling salesman 4-opt move in the same time as a best 2-opt move

A special class of 4-opt moves plays a key role in several leading heuristics for the traveling salesman problem (TSP). However, the number of such moves is quite large—O(n ⁴) for a graph of n nodes, on the order of the square of the number of 2-opt moves. Consequently, classical TSP heuristics have not attempted to seek best (and often not even improving) instances of these moves. We show that a best move from the collection that consists of these moves, together with an additional class of 4-opt moves and certain related 3-opt moves, can nevertheless be found in the same order of time required to find a best 2-opt move. Our method employs an acyclic shortest path model based on ideas introduced with ejection chain procedures and generates a sequence that can include improving moves at earlier stages. Joined with candidate list strategies that limit the tour edges available to be dropped, the method can also be structured to find best members from the set of implied surviving moves in O(n) time, making available TSP strategies for incorporating 4-opt moves that were previously beyond practical consideration.     

The <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript> regularity of a class of non-homogeneous ultraparabolic equations

We obtain the C ^α regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent C ^α regularity results of homogeneous ultraparabolic equations. This work was supported by National Natural Science Foundation of China (Grant No. 10325104)     

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth-order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L^∞ Kähler metric. The main result is to show that such a weak solution (with uniform L^∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K-energy minimizers. The new technical ingredient is a W^{2, 2} regularity result for the Laplacian equation Δ_gu=f on Kähler manifolds, where the metric has only L^∞ coefficients. It is well-known that such a W^{2, 2} regularity (W^{2, p} regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form for a^ij ∊ L^∞ without certain smallness assumptions on the local oscillation of a^ij. We observe that the Kähler condition plays an essential role in obtaining a W^{2, 2} regularity for elliptic equations with only L^∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.     

Sous ellipticite dans le cadre du calcul S(m,g)

Abstract

We study the regularity of the free boundary in the two membranes problem. We prove that around any point the free boundary is either a C ^{1, α} surface or a cusp, as in the obstacle problem. We also prove C ^{1, 1} regularity for the pair of functions solving the problem.     

Partial regularity for the Landau-Lifshitz system

The aim of this work is to consider the partial regularity for the stationary weak solutions to the Landau-Lifshitz system of Ferromagnetic spin chain from a m-dimensional manifold M into the unit sphere S² of R³. The Landau-Lifshitz system is in appearance very similar to the heat flows of harmonic maps into sphere. However the monotonicity inequality, which plays an important role in getting partial regularity, does not hold in this case. This becomes a large barrier to regularity. In the present paper we get a generalized monotonicity inequality, and find the singular set of the stationary weak solutions of Landau-Lifshitz system.Received: 23 December 2002, Accepted: 10 July 2003, Published online: 22 September 2003Mathematics Subject Classification (1991): 58E20, 58Z05Project 10071013 supported by NSFC.     

Optimal <Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript> regularity theory for parabolic equations in divergence form

This work treats L^p regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W^1,p regularity theory holds. This work was supported by SNU foundation in 2005.     

Potential theory for regular and mach reflection of a shock at a wedge

If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 2θ_w through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter β = c₁θ²_w(γ + 1)b ranges from 0 to ∞. Here γ is the polytropic constant and C₁ is the sound speed behind the incident shock. For β > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For β < 1/2 Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case β = 0 can be solved. For 1/2 < β < 2 or even larger β the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens. The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for β sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point. © 1994 John Wiley & Sons, Inc.     

Non‐standard Stokes and Navier–Stokes problems: existence and regularity in stationary case

This paper is devoted to Stokes and Navier–Stokes problems with non‐standard boundary conditions: we consider, in particular, the case where the pressure is given on a part of the boundary. These problems were studied by Bégue, Conca, Murat and Pironneau. They proved the existence of variational solutions, indicating that these were solutions of the initial non‐standard problems, if they are regular enough, but without specifying the conditions on the data which would imply this regularity. In this paper, first we show that the variational solutions, on supposing pressure on the boundary Γ₂ of regularity H^1/2 instead of H^?1/2, have their Laplacians in L² and, therefore, are solutions of non‐standard Stokes problem. Next, we give a result of regularity H², which we generalize, obtaining regularities W^{m, r}, m∈?, m?2, r?2. Finally, by a fixed‐point argument, we prove analogous results for the Navier–Stokes problem, in the case where the viscosity νis large compared to the data. Copyright © 2002 John Wiley & Sons, Ltd.     

Second-Order Elliptic Equations of Nondivergence Form with Small BMO Coefficients in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper we obtain the global regularity estimates in Orlicz spaces for second-order elliptic equations of nondivergence form with small BMO coefficients in ℝ ⁿ . As a corollary we obtain L ^p -type regularity for such equations. Our results improve the known results for such problems.     

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Let X* be a subset of an affine space 𝔸 ^s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ? ^s?1 and ? ^s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.     

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels

We prove a C ^{1, α} interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.     

Generating uniformly distributed random latin squares

By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of n × n Latin squares, we can obtain squares that are (approximately) uniformly distributed; we offer two such chains. The central issue is the construction of “moves” that connect the squares. Our first approach uses the fact that an n × n Latin square is equivalent to an n × n × n contingency table in which each line sum equals 1. We relax the nonnegativity condition on the table's cells, allowing “improper” tables that have a single—1-cell. A simple set of moves connects this expanded space of tables [the diameter of the associated graph is bounded by 2(n − 1)³], and suggests a Markov chain whose subchain of proper tables has the desired uniform stationary distribution (with an average of approximately n steps between proper tables). By grouping these moves appropriately, we derive a class of moves that stay within the space of proper Latin squares [with graph diameter bounded by 4(n − 1)²]; these may also be used to form a suitable Markov chain. © 1996 John Wiley & Sons, Inc.     

C ∞ regularity of solutions of an equation of Levi’s type in R 2 n +1

We study the regularity of the solutions u of a class of P.D.E., whose prototype is the prescribed Levi curvature equation in ℝ^{2 n +1}. It is a second-order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every function u∈C ². If the Levi curvature never vanishes, we represent the operator ℒ associated with the Levi equation as a sum of squares of non-linear vector fields which are linearly independent at every point. By using a freezing method we first study the regularity properties of the solutions of a linear operator, which has the same structure as ℒ. Then we apply these results to the classical solutions of the equation, and prove their C ^∞ regularity. Received: October 10, 1998; in final form: March 5, 1999?Published online: May 10, 2001