Existence of solutions for a nonisothermal Allen-Cahn type system is proved by using a semidiscrete spectral Galerkin method together with degree theory and maximum principle. Regularity and uniqueness are obtained in a special situation for domains with dimension up to two and smooth enough data. The present system may model the evolution of solidification or melting processes occurring in certain binary alloys.the limit to obtain solutions of the original problem. Regularity and uniqueness are obtained for the case of two-dimensional domains when k D k.'/, D 1 D D 1 .', Â/ and D 2 D D 2 .', Â/, and the other data are smooth enough. In this special case, error estimates for the semidiscrete approximations can be obtained. The precise statements of such results are presented in the next section.We remark that for simplicity of exposition, we assumed homogeneous Dirichlet boundary conditions. With simple modifications of the arguments, similar analysis could be performed for other kinds of boundary conditions; for instance, we could take homogeneous Neumann boundary conditions for some of all of the unknown or consider appropriate nonhomogeneous boundary conditions. Also for simplicity, we presented the potential nonlinearity in the phase-field equation (1.1) as '.' 1/.1 2'/, that is, the one derived from the two-well potential. Our results also hold, with the same proofs, to more general nonlinearities as the ones in class presented in [7].This work is organized as follows. In Section 2, we set up the notations and the functional spaces used in the paper; we recall certain interpolation and embedding results; we also explicitly state the technical hypotheses to be used and our main results . In Section 3, we consider the existence and uniqueness of solutions for two auxiliary phase-field equations. One of them was already treated by Moroşanu and Motreanu [7], and for convenience of reference, we state their result here. This result will be used to investigate the solvability of the second phase-field equation, which is directly related to the phase-field equation (1.1) present in our problem. For this, we will use a suitable maximum principle, regularization, and Leray-Schauder degree theory. Existence of a weak solution is proved in Section 4 with the use of a semidiscrete spectral Galerkin method. Section 5 is dedicated to regularity results presented in the onedimensional and two-dimensional situations. For the special case when k D k.'/, D 1 D D 1 .', Â/ and D 2 D D 2 .', Â/, we obtain further regularity and uniqueness of the solutions.Concerning notations, as it is usual in this kind of context, throughout the article we will denote by M, and sometimes M 1 , M 2 , : : :, constants depending only on known quantities. Also, to shorten the presentation, the proofs of our results will be performed in the highest dimension being considered in each case; the proofs for lower dimensions are easier.
A mathematical analysis of a phase-field model for solidification non-isothermal of a binary alloy with convection is presented. Convergence of the solutions of discretized scheme is proved and existence result for original problem are derived.
Este artigo é um recorte da dissertação de mestrado Atos e Lugares de Aprendizagem Criativa em Matemática. O principal objetivo é apresentar como o processo de impressão tridimensional pode potencializar ações interdisciplinares para promover uma aprendizagem criativa em Matemática. Tais ações são norteadas pelos princípios da Cultura Maker (“aprender fazendo”) e da metodologia STEAM (acrônimo formado pelas iniciais dos nomes, em inglês, das disciplinas ciências, tecnologia, engenharia, arte e matemática), realizadas no lugar de aprendizagem criativa, denominado Garagem, que tem como proposta experimentar uma “matemática mão na massa” por meio da prototipagem de objetos de aprendizagem. Como metodologia de pesquisa, utiliza-se o método da cartografia, ancorado na proposta dos filósofos Gilles Deleuze e Félix Guattari. O conceito de aprendizagem criativa tem como referenciais teóricos as ideias de aprendizagem defendidas pelo educador Paulo Freire; o conceito de criatividade, por sua vez, segundo o psicanalista Donald Winnicott; por interdisciplinaridade, nos apoiamos nas ideias de Ivani Fazenda. Os resultados da pesquisa apontaram que a impressão 3D promove uma aprendizagem criativa, pois valoriza um processo interdisciplinar em que o aluno é o protagonista, permitindo-lhe (re)criar saberes de modo próprio e original, de modo a possibilitar uma aprendizagem mais autônoma, autoral e criativa.
In this article, under certain conditions, we prove the regularity for the solutions of an Allen-Cahn phase-field type system obtained as limits of approximate solutions constructed by using a semidiscrete spectral Galerkin method. With the help of this improved regularity, as one compares to previous results, we then derive error estimates for the approximate solutions in terms of the inverse of the eigenvalues of the Laplacian operator. The system under investigation may model the evolution of solidification or melting of certain binary alloys.
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