Generalized solutions in PDEs and the Burgers' equation

In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level

Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterising their combinatorial properties by means of their monads. This extends to arbitrary sets and properties methods previously used to study partition regular Diophantine equations on N. Several applications are described by means of multiple examples

Partition regulairty of nonlinear polynomials: a nonstandard approach

In 2011, Neil Hindman proved that for all natural numbers n, m the polynomial Pn i=1 xi Q m j=1 yj has monochromatic solutions for every finite coloration of N. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials P(x1, . . . , xn, y1, . . . , ym) of the following kind: P(x1, . . . , xn, y1, . . . , ym) = Pn i=1 aixiMi(y1, . . . , ym), where n, m are natural numbers, Pn i=1 aixi has monochromatic solutions for every finite coloration of N and the degree of each variable y1, . . . , ym in Mi(y1, . . . , ym) is at most one. An example of such a polynomial is x1y1 + x2y1y2 x3. The second class of polynomials generalizing Hindman\u2019s result is more complicated to describe; its particularity is that the degree of some of the involved variables can be greater than one. The technique that we use relies on an approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most interesting aspect of this technique is that, by carefully choosing the appropriate nonstandard setting, the proof of the main results can be obtained by very simple algebraic considerations

Asymptotic gauges: Generalization of Colombeau type algebras

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general "growth condition" formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved

A model problem for ultrafunctions

IIn this article. we show that non-Archimedean mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some physical problems which cannot be described by the usual mathematics. The problem which we will consider here is the minimization of the functional E(u, q) = 1/2 integral(Omega)vertical bar del u(x)vertical bar(2) dx + u(q). When Omega subset of R-N is a bounded open set and u is an element of C-0(2) (Omega), this problem has no solution since inf E (u, q) = -infinity. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply the general ideas of NAM and some of the techniques of Non Standard Analysis to a new notion of generalized functions, called ultrafunctions, which are a particular class of functions based on a Non-Archimedean field. In this class of functions, the above problem is well posed and it has a solution

Generalized solutions of variational problems and applications

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach

The classical theory of calculus of variations for generalized functions

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

A model problem for ultrafunctions

IIn this article. we show that non-Archimedean mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some physical problems which cannot be described by the usual mathematics. The problem which we will consider here is the minimization of the functional E(u, q) = 1/2 integral(Omega)vertical bar del u(x)vertical bar(2) dx + u(q). When Omega subset of R-N is a bounded open set and u is an element of C-0(2) (Omega), this problem has no solution since inf E (u, q) = -infinity. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply the general ideas of NAM and some of the techniques of Non Standard Analysis to a new notion of generalized functions, called ultrafunctions, which are a particular class of functions based on a Non-Archimedean field. In this class of functions, the above problem is well posed and it has a solution