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Weak Reproductive Solutions for a Convection-Diffusion Model Describing a Binary Alloy Solidification Processes

Blanca Climent-Ezquerra, Mario Durán, Elva Ortega-Torres, Marko Rojas-Medar

Abstract

We study the existence of reproductive weak solutions for a system of equations describing a solidification process of a binary alloy confined into a bounded and regular domain in $\mathbb{R}^3$, having mixed boundary conditions.

Weak Reproductive Solutions for a Convection-Diffusion Model Describing a Binary Alloy Solidification Processes

Abstract

We study the existence of reproductive weak solutions for a system of equations describing a solidification process of a binary alloy confined into a bounded and regular domain in , having mixed boundary conditions.
Paper Structure (9 sections, 11 theorems, 100 equations)

This paper contains 9 sections, 11 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Mathematical model for a binary alloy solidification
  3. The physical problem
  4. Solidification model for a binary alloy
  5. Setting the evolutive mathematical model
  6. Function spaces and preliminaries
  7. The regularized mathematical problem
  8. Weak reproductive solution of the regularized problem
  9. Reproductive weak solution of the solidification problem

Key Result

Lemma 3.1

(Young's Inequality) Let $A\geq 0, \, B\geq 0$ be real constants. Suppose that $p, \, q \in \mathbb{R}$ such that $\, 1\leq p < \infty\,$ and $\, \frac{1}{p} + \frac{1}{q} = 1$. Then, for all $\, \xi > 0\,$, the following inequality hold:

Theorems & Definitions (16)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Definition 3.4
  • Proposition 3.5
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1
  • Remark 5.2
  • ...and 6 more