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Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law

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Chinese Annals of mathematics,Series B 2023年第4期44卷 561-576页

作者： Aifang QU Hairong YUAN Department of mathematics Shanghai Normal UniversityShanghai 200234China School of mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice East China Normal UniversityShanghai 200241China

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space,the shock-front coincides with the upwind surface of the body,hence there is an infinite-thin layer of concentrated mass,in which all particles hitting the body move along its upwind surface.By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations,which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces,the authors derive the compressible Euler equations for flows in concentration layers,which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface.As a by-product,they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows,which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.

关键词： Compressible Euler equations Hypersonic fow Concentration layer Ramp Cone Radon measure solution Newton-Busemann law

THE GLOBAL LIPSCHITZ SOLUTION FOR A PEELING MODEL

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数学物理学报:B辑英文版 2023年第5期43卷 2263-2278页

作者：黎前锋张永前 School of mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice East China Normal UniversityShanghai 200241China School of mathematical Sciences Fudan UniversityShanghai 200433China

This paper focusses on a peeling phenomenon governed by a nonlinear wave equation with a free boundary.Under the hypotheses that the total variation of the intial data and the boundary data are small,the global existence of a weak solution to the nonlinear problem(1.1)-(1.3)is proven by a modified Glimm scheme.The regularity of the peeling front is established,and the asymptotic behaviour of the obtained solution and the peeling front at infinity is also studied.

关键词： peeling model nonlinear wave solution free boundary Glimm scheme

ON FINITE ELEMENT APPROXIMATIONS TO A SHAPE GRADIENT FLOW IN SHAPE OPTIMIZATION OF ELLIPTIC PROBLEMS

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Journal of Computational mathematics 2023年第5期41卷 956-979页

作者： Chunxiao Liu Shengfeng Zhu School of Statistics and mathematics Shanghai Lixin University of Accounting and FinanceShanghai 201209China Department of mathematics&Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice East China Normal UniversityShanghai 200241China

Shape gradient flows are widely used in numerical shape optimization algorithms.We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems.We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative.Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.

关键词： Shape optimization Shape gradient Eulerian derivative Finite element Error estimate

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

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Acta Mathematica Scientia 2023年第3期43卷 1045-1080页

作者：温丽丽范恩贵陈勇 School of mathematical Sciences Shanghai Key Laboratory of Pure Mathematics and Mathematical Practiceand Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghai200241China School of mathematical Sciences and key laboratory for Nonlinear Science Fudan UniversityShanghai200433China College of mathematics and Systems Science Shandong University of Science and TechnologyQingdao266590China

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

关键词： Sasa-Satsuma equation nonzero boundary condition auxiliary eigenfunctions Riemann-Hilbert problem soliton solution

Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay

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Chinese Annals of mathematics,Series B 2016年第2期37卷 291-308页

作者： Xianlong FU Jialin ZHANG Department of mathematics Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice East China Normal University

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.

关键词：逼近能控性微分系统线性泛函中立型状态时滞功率理论空间变量半群理论

Integrability and Exact Solutions of the(2+1)-dimensional KdV Equation with Bell Polynomials Approach

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Acta Mathematicae Applicatae Sinica 2022年第4期38卷 861-881页

作者： Jun-cai PU Yong CHEN School of mathematical Sciences Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghai 200062China College of mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China

In this paper,the bilinear formalism,bilinear B?cklund transformations and Lax pair of the(2+1)-dimensional KdV equation are constructed by the Bell polynomials approach.The N-soliton solution is derived directly from the bilinear form.Especially,based on the two-soliton solution,the lump solution is given out analytically by taking special parameters and using Taylor expansion formula.With the help of the multidimensional Riemann theta function,multiperiodic(quasiperiodic)wave solutions for the(2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method.Moreover,the asymptotic properties of the one-and two-periodic wave solution,which reveal the relations with the single and two-soliton solution,are presented in detail.

关键词： the bilinear formalism bilinear B?cklund transformations Lax pair lump solution periodic wave solution the asymptotic properties

Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation

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Chinese Annals of mathematics,Series B 2023年第1期44卷 81-98页

作者： Qian YANG Mingkang NI College of Science University of Shanghai for Science and TechnologyShanghai 200093China School of mathematical Sciences East China Normal UniversityShanghai 200062China Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice Shanghai 200241China

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied.A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered.The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied.An efficient algorithm for constructing the point itself and an asymptotic representation of arbitrary-order accuracy to the solution is proposed.For sufficiently small parameter values,the existence theorem is proved by the technique of matching asymptotic expansions.An example is given to show the effectiveness of their method.

关键词： Reaction-Diffusion-Advection equation Internal layer Asymptotic method Piecewise-Smooth dynamical system

LIPSCHITZ STAR BODIES

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Acta Mathematica Scientia 2023年第2期43卷 597-607页

作者：蔺友江吴尉迟 School of mathematics and Statistics Chongqing Technology and Business UniversityChongqing 400067China School of mathematics science East China Normal UniversityShanghai 200241China Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice Shanghai 200241China

In this paper,we study some basic properties on Lipschitz star bodies,such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball,the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance,and the convergence of Steiner symmetrizations of Lipschitz star bodies.

关键词： Steiner symmetrization star body Lipschitz function

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

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Communications on Applied mathematics and Computation 2023年第3期5卷 1097-1129页

作者： Yunjuan Jin Aifang Qu Hairong Yuan Center for Partial Differential Equations School of Mathematical SciencesEast China Normal UniversityShanghai200241China Department of mathematics Shanghai Normal UniversityShanghai 200234China School of mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice East China Normal UniversityShanghai 200241China

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

关键词： Compressible Euler equations Radon measure solution Delta shock Riemann problem Non-uniqueness