# On the physics background of p-Laplacian equation

Could you tell me the physics background of p-Laplacian equation? Thank you! Actually, I know nothing about this. But I am curious about the original of these PDEs or where they come from. Could you introduce some related books or articles to me?

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I have to agree with Carnahan. It's rather hard answering this if we don't know how much background in physics/engineering you have. I tried posting an answer that should satisfy an undergrad physics/engineering student. If you need more clarification, I'd gladly give more insight. – Michael Kissner May 30 '11 at 7:20

It is used to model Newtonian and non Newtonian Fluids. You generally have different values for $p$ depending on the type of fluid:

• $p=2$ (Newtonian) for fluids like water and air
• $p>2$ for viscous ("sticky") fluids
• $p<2$ for non-viscous ("runny") Fluids

Edit: I just realized that this alone probably doesn't satisfy your question. Let me clearify the actual physics behind the above thinking:

The viscous property of those fluids arises from shear stress ("drag") inside of the fluid. To describe this effect in fluids, we use the speed the particles travel in relation to each other (the factor being $\vert \nabla u \vert ^{p-2}$ ).

Newtonian fluids are purely linear, so the correct approximation is $p=2$. For non-newtonian the viscosity is non-linear, so $p\not= 2$.

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