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Related Concept Videos

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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...
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The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).
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Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...
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Phase Changes01:19

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Phase transitions play an important theoretical and practical role in the study of heat flow. In melting or fusion, a solid turns into a liquid; the opposite process is freezing. In evaporation, a liquid turns into a gas; the opposite process is condensation.
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When a substance—isolated from its environment—is subjected to heat changes, corresponding changes in temperature and phase of the substance is observed; this is graphically represented by heating and cooling curves.
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The internal energy of a substance—the total kinetic energy of all its molecules and the potential energy of their associated forces—depends on the strength of the intermolecular forces in the condensed phases and the pressure exerted on the substance. The internal energy of a substance is the highest in the gaseous state, the lowest in the solid state, and intermediate in the liquid state. Phase transitions are caused by changes in physical conditions, such as temperature and...
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Continuity of the temperature in a multi-phase transition problem.

Ugo Gianazza1, Naian Liao2

  • 1Dipartimento di Matematica "F. Casorati", Università di Pavia, via Ferrata 5, 27100 Pavia, Italy.

Mathematische Annalen
|September 7, 2022
PubMed
Summary

Weak solutions for a doubly nonlinear parabolic equation, modeling material phase transitions, are proven to be locally continuous. An explicit modulus of continuity is provided, considering p-Laplacian diffusion effects.

Keywords:
35B6535K6535K9280A22

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Area of Science:

  • Partial Differential Equations
  • Materials Science
  • Continuum Mechanics

Background:

  • Doubly nonlinear parabolic equations model complex phenomena like multi-phase transitions in materials.
  • Understanding the regularity of weak solutions is crucial for analyzing material behavior.
  • The p-Laplacian operator introduces nonlinear diffusion, affecting solution properties.

Purpose of the Study:

  • To establish the local continuity of weak solutions for a specific doubly nonlinear parabolic equation.
  • To derive an explicit modulus of continuity for these solutions.
  • To investigate the influence of p-Laplacian type diffusion on solution regularity.

Main Methods:

  • Analysis of weak solutions in appropriate function spaces.
  • Application of techniques for proving regularity of solutions to nonlinear PDEs.
  • Detailed examination of the properties of the p-Laplacian operator in the diffusion term.

Main Results:

  • Local weak solutions are demonstrated to be locally continuous.
  • An explicit mathematical expression for the modulus of continuity has been derived.
  • The impact of p-Laplacian diffusion on the continuity of solutions has been analyzed.

Conclusions:

  • The local continuity of weak solutions provides a significant regularity result for this class of equations.
  • The derived modulus of continuity offers quantitative insights into solution behavior.
  • The study contributes to the mathematical understanding of phase transitions in materials governed by nonlinear diffusion.