Equations Of Mixed Elliptic-hyperbolic Type Research Articles

Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type

This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.

On Some Boundary Value Problems with a Shift for a Mixed Type Equation

Важным этапом в становлении теории краевых задач стали предложенные А.М. Нахушевым в 1969 году нелокальные задачи нового типа, впоследствии названные у нас краевыми задачами со смещением, а за рубежом — задачами (проблемами) Нахушева. Они являются обобщением задачи Трикоми, а так же содержат широкий класс корректных самосопряженных задач. Эти задачи сразу вызвали большой интерес многих авторов. За последние годы исследования задач со смещением для уравнений смешанного типа ведутся особенно интенсивно. Но в этих работах краевые условия, как правило, содержат классические операторы, в то время как нелокальным краевым задачам, содержащим операторы более сложной структуры и операторы дробного интегро-дифференцирования. Настоящая статья посвящена исследованию вопроса однозначной разрешимости краевых задач со смещением для уравнения смешанного эллиптико-гиперболического типа. Сформулированы корректные краевые задачи со смещением для уравнения смешанного типа. В данной работе исследованы вопросы однозначной разрешимости задач со смешением для уравнения смешанного эллиптико-гиперболического типа в смешанной области Ω, ограниченной в полуплоскости y>0 гладкой кривой Жордана, а в полуплоскости y<0 характеристиками уравнения (1). При ограничениях неравенственного типа на известные функции и различных порядках операторов дробного дифференцирования в краевом условии доказаны теоремы единственности. Существование решения задач доказывается путем редукции задач к уравнениям Фредгольма второго рода, безусловная разрешимость которых следует из единственности решения задач. An important stage in the development of the theory of boundary value problems was the proposed by A.M. Nakhushev in 1969, non-local problems of a new type, which were later called in our country boundary value problems with a shift, and abroad – Nakhushev problems (problems). They are a generalization of the Tricomi problem, and also contain a wide class of well-posed self-adjoint problems. These problems immediately aroused great interest of many authors. In recent years, studies of problems with a shift for equations of mixed type have been carried out especially intensively. But in these works, the boundary conditions, as a rule, contain classical operators, while non-local boundary value problems contain operators of a more complex structure and operators of fractional integro-differentiation. In this paper, we study the unique solvability of problems with mixing for an equation of mixed elliptic-hyperbolic type. Under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition, uniqueness theorems are proved. The existence of a solution to the problems is proved by reducing the problems to Fredholm equations of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problems.

ОБ АНАЛОГЕ ЗАДАЧИ ТРИКОМИ ДЛЯ УРАВНЕНИЯ ТРЕТЬЕГО ПОРЯДКА СМЕШАННОГО ТИПА

It is commonly known that equations of mixed type are partial differential equations that belong to different types in different parts of the domain under consideration. For example, the equation may belong to the elliptic type in one part of the domain and to the hyperbolic type in another one; these parts are separated by a transition line, at which the equation degenerates into parabolic or undefined. In 1923, the Italian mathematician F. Tricomi considered a boundary value problem for one equation of mixed elliptic-hyperbolic type (later named after him) in a domain bounded in the upper halfplane by the Lyapunov curve, and in the lower half-plane by the characteristics of the equation that emerge from the ends of this curve; the boundary conditions were then set on the curve and on one of the characteristics. The solution had to be continuous in the closure of the domain, continuously differentiable within it, and twice continuously differentiable in the upper (elliptic) and lower (hyperbolic) subdomains; for the first derivatives of the solution, singularities of integrable order were allowed near the ends of the curve. Tricomi proved the existence and uniqueness of the solution to the problem in the specified class; when proving the existence, he reduced the problem to a singular integral equation. The article studies an analogue of the Tricomi problem for a third-order hyperbolic-parabolic equation of mixed type with a spectral parameter. The uniqueness and existence of a solution to the problem are proved. The uniqueness of the solution to the problem is proved by the method of energy integrals, and the existence of the solution is proved by the method of reduction to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the solution to the problem.

The First Boundary Problem for a Mixed Type Equation with Characteristic Degeneracy with Discontinuous Frankl Condition

For an equation of mixed elliptic-hyperbolic type of the second kind, the boundary problem in a rectangular region is studied. The criterion of uniqueness is established. The solution is built as a sum row. When substantiating convergence, the problem of small denominators. In this connection, estimates of separation from zero small denominators with the corresponding asymptotics, which and allowed us to justify the existence of a solution in the class of regular solutions and solution stability from boundary data.

The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

Nonlocal Inverse Problem of Finding Unknown Multipliers in the Right-Hand Part of Lavrentiev-Bitsadze Equation

We consider an equation of mixed elliptic-hyperbolic type. The right-hand part of the equation is represented as a product of two functions, each which only depends on one of variables. We study an inverse problem for this equation; it consists in finding the unknown multipliers. We establish a uniqueness criterion for this problem. The solution is constructed as a series in eigenfunctions for the corresponding one-dimensional spectral problem. Under some conditions on the boundary of domain, boundary values, and the unknown multipliers, we establish separation from zero of small denominators of ratios containing in the coefficients of the series; the existence of solution and its stability are proved as well.

A. A. Dezin’s problem for inhomogeneous Lavrent’ev–Bitsadze equation

We establish a uniqueness criterion for solution of nonlocal Dezin’s problem for an equation of mixed elliptic-hyperbolic type. The solution is constructed as a sum of a series in eigenfunctions of the corresponding one-dimensional spectral problem. In the proof of its convergence there arises a problem on small denominators arises. Under certain restrictions on the given parameters and functions we prove the convergence of constructed series in the class of regular solutions.

The inverse problem for the Lavrent’ev–Bitsadze equation connected with the search of elements in the right-hand side

We consider an equation of mixed elliptic-hyperbolic type, whose right-hand side represents a product of two one-dimensional functions. We establish a criterion for the unique solvability of this equation and construct its solution as a sum of series on the set of its eigenfunctions. Under certain constraints imposed on the ratio of the rectangle sides, on boundary functions, and on known multipliers in the right-hand side of the equation, we obtain estimates separating small denominators that appear in coefficients of constructed series from zero.

Solutions of fixed sign of higher-order inhomogeneous equations of mixed elliptic-hyperbolic type

For a higher-order inhomogeneous equation of mixed elliptic-hyperbolic type, the property of the solution to be of fixed sign is established, depending on the sign of the right-hand side.

Нелокальная задача А. А. Дезина для уравнения смешанного эллиптико-гиперболического типа

В данной работе для уравнения смешанного эллиптико-гиперболического типа в прямоугольной области изучена задача с условиями периодичности и нелокальным условием А. А. Дезина. Установлен критерий единственности. Решение задачи построено в виде суммы ортогонального ряда по собственным функциям соответствующей одномерной спектральной задачи. При обосновании сходимости ряда возникает проблема малых знаменателей. В связи с этим установлена оценка отделенности от нуля малых знаменателей с соответствующей асимптотикой. Эта оценка позволила при некоторых условиях относительно заданных параметров задачи и функций доказать сходимость построенного ряда в классе регулярных решений и устойчивость решения.

The dirichlet problem for an equation of mixed elliptic-hyperbolic type with variable potential

We study the Dirichlet problem for an equation of mixed elliptic-hyperbolic type with variable potential in the rectangular area. We establish a criterion of the uniqueness of a solution to this problem. The uniqueness of a solution is proved on the basis of the completeness of systems of functions corresponding to one-dimensional spectral problem. Solutionwas constructed as a sumof series on the systems of eigenfunctions. The existence is proved under certain conditions on the ratio of the rectangle sides of hyperbolic part of the equation, upon the boundary functions and function of potential.

On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas

For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.

Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System

We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow, through the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition. We establish a global theory of existence and optimal regularity for this shock diffraction problem. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.

Problem with shifts in interior characteristics for equations of mixed elliptic-hyperbolic type with a singular coefficient

We consider a problem with local shift conditions on a segment of the degeneration line and with shifts in interior characteristics of the equation. The uniqueness of the solution is proved with the use of the extremum principle. To prove the existence of a solution, we apply the theory of singular integral equations and Fredholm integral equations.

On the Dirichlet problem for a mixed-type equation of the second kind with strong degeneration

By separation of variables, we construct particular solutions of an equation of mixed elliptic-hyperbolic type of the second kind with strong degeneration in a rectangular domain. We pose certain transmission conditions on the singular line. The constructed solutions satisfy the equation in the entire mixed domain and are used to study the uniqueness of the solution of the Dirichlet problem for this domain.

Thoughts on the Busemann equation

A second-order quasi-linear partial differential equation of mixed elliptic-hyperbolic type in two independent variables, which mimics one introduced by A. Busemann in gas dynamics, and arises in the study of Minkowski spaces and in other theories, is considered.

Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

AbstractWe establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C1,α, provided that the hyperbolic phase is close in C1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C2,α, the free boundary is C2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.

Some recent methods for partial differential equations of divergence form

Some recent methods for solving second-order nonlinear partial differential equations of divergence form and related nonlinear problems are surveyed. These methods include entropy methods via the theory of divergence-measure fields for hyperbolic conservation laws, kinetic methods via kinetic formulations for degenerate parabolichyperbolic equations, and free-boundary methods via free-boundary iterations for multidimensional transonic shocks for nonlinear equation of mixed elliptic-hyperbolic type. Some recent trends in this direction are also discussed.

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

We establish 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 a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.