Energy in the heat equation. - Mathematics Stack Exchange
Feb 8, 2015 · Good question. The energy I would associate with the heat equation is the Dirichlet energy, $\int u_x^2.$
heat equation conservation of energy - Mathematics Stack Exchange
May 4, 2016 · so that thet total heat energy is conserved: $$\int\limits_ {-\infty}^ {\infty}u (x,t)dx = \text { constant.} \quad (2)$$ Could someone explain how to get from the initial conditions to …
functional analysis - Duhamel's principle for heat equation ...
Jan 18, 2018 · functional-analysis partial-differential-equations sobolev-spaces convolution heat-equation See similar questions with these tags.
Is every classical solution of the inhomogeneous heat equation …
Sep 19, 2025 · So does he talk about the solution defined by the Duhamel formula? Remarks To check that the Duhamel formula is a solution one just needs to differentiate once with respect …
sobolev spaces - Heat equation: proving that smaller diffusion …
Heat equation: proving that smaller diffusion leads to bigger solution via energy methods Ask Question Asked 2 years, 5 months ago Modified 2 years, 5 months ago
partial differential equations - How to you prove that the Heat …
Aug 4, 2021 · How to you prove that the Heat Kernel of the Sub-Laplacian is given by this formula? Ask Question Asked 4 years, 1 month ago Modified 4 years, 1 month ago
Two questions about Lemma 7.15 of the Localization formula in …
Sep 15, 2025 · 4 I am working through the proof of the Berline-Vergne localization formula (to be specific in Lemma 7.15 in Heat Kernels and Dirac Operators) and I have two questions, the …
Total energy of a pendulum - Mathematics Stack Exchange
May 27, 2023 · Your potential energy is the actual energy as it is the potential energy at the extreme where kinetic energy is $0$. You have added maximum potential energy and …
Understanding the Laplace operator conceptually
Mar 17, 2021 · This insight leads to the heat equation $$\frac {df} {dt} = \Delta f$$ which, given initial temperatures on $\Omega$, flows in the direction that best decreases the Dirichlet …
real analysis - What are all $L^pL^q$ estimates for the heat …
Mar 20, 2023 · Basically, the problem is to find all the 'Strichartz estimates' of the heat equation with fractional gain of derivatives. Note that this is not about the fractional heat equation: we …