On a free boundary problem for the Navier-Stokes equations

Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small‐amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small‐amplitude time‐periodic solutions.

Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face

SynopsisWe consider the problemwhere H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove:(i) There is a smooth initial function u0, u0≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u0.(ii) The comparison principle: The relation for any x ≠ 0, implies u1(t)>u2(t), t≧0 for any u1, u2 solving the problem with the initial data , respectively.(iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {(x, t)| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation $$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$ wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.

Existence and uniqueness theorems are established for initials-boundary value problems corresponding to the equaction under assumptions on P and a that include as special cases the isentropic gas law P(v) = v−γ with γ>1 with α(v) = v −1 along with generalizations that allow for Non convex P. The smoothness assumptions made on P and α are much weaker then are usual in studying these problems and the initial data is only required to satisfy .

SYNOPSISThis paper extends some recent results of V. Barbu and H. Brézis. It is concerned with bounded solutions of the problem pu″+qu′ ∈ Au, u(0) = a, where A is a maximal monotone operator in a real Hilbert space H and p and q are real functions. Existence and uniqueness theorems are proved, with results on integrability of solutions in various measure spaces on R+. T(t) denotes the family of contractions of D(A) generated by the equation and we obtain a regularising effect on the initial data. Some properties of this family of contractions are studied.

The inverse problems of finding the free term and the coefficient of u(x,t) in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse problem and the completeness of a certain system of functions is established.

An iterative scaling function procedure for solving scalar non-linear hyperbolic balance laws

A free boundary problem for the nonhomogeneous wave equation is studied. Such a problem arises when the motion of a load on a string is analyzed without supposing that the load velocity is known and fixed. Preliminarily the complementary equations which characterize the free boundary are determined. The mechanical problem is then solved by proving a uniqueness and existence theorem.

Background : The teaching of differential equations is dominated by an excessively algebraised analytic tradition. For this reason, studies that contribute to conceptualising mathematical objects associated with the differential equation are important, particularly the existence and uniqueness theorem. Objectives : From its genesis, the objective is to analyse the nature of this knowledge, its epistemology from practice. We give an account of the variational arguments , i.e. , based on practices focused on the study of change, with a predictive purpose, which allows obtaining the desired result on the differential equation: demonstrating the existence of a unique solution. Design : A documentary analysis is carried out from the Socioepistemological Theory of the works that marked the construction of this mathematical knowledge. Setting and Participants : Being a documentary-cut study, we did not have participants stricto sensu . Data collection and analysis : Our observation unit includes mathematical works as primary and secondary sources involved in constructing the theorem: its postulations, search for hypotheses and proofs. Results : A reconstruction of the theorem is offered, which from the arguments, characterises some practices that helped in the construction of mathematical objects. Conclusions : We conclude that the bounded variation, as a particular way of using change, contributed to the search or establishment of conditions for the interpretation of the solution of equations and to obtain a unique solution to the differential equation, contributions that should be key for implementations of learning situations.

It is known that in order for a solution of a boundary value problems for elliptic equations in unbounded domains to be unique, it is necessary to impose additional constraints on the behavior, of solutions at infinity. Such constrains may be qualitatively different.In [1] it was noticed that the assumption that the solutions should be positive for sufficiently large arguments may also serve as a condition that guarantees the uniqueness of solutions. The present report explains an investigation of the conditions on boundary value problems whose fulfillment guarantees the validity of such uniqueness theorems for the biharmonic equation. For harmonic functions that satisfy the Dirichlet or Neumann boundary conditions, there are no such uniqueness theorems. But they can exist for simplest nonlocal boundary value problems for harmonic functions.

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to $\mathbb{R}^3$. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to $\mathbb{R}^3$ contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to $\mathbb{R}^3$ minus a finite number of open balls. An extension to higher dimensions is also discussed.