Ross Ogilvie      11th November, 2024
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33.

The distribution of heat

Consider the fundamental solution of the heat equation $\mathrm{\Phi }(x,t)$ given in Definition 4.5.

(a)

Show that this extends to a smooth function on ${ℝ}^n\times ℝ\setminus \{(0,0)\}$. (2 points)

(b)

Verify that this obeys the heat equation on ${ℝ}^n\times ℝ\setminus \{(0,0)\}$. (2 points)

We want to show that $\phi \mapsto H(\phi )={\int }_{{ℝ}^n\times ℝ}\mathrm{\Phi }(x,t)\phi (x,t)\mathit{𝑑𝑥}\mathit{𝑑𝑡}$ is a distribution. Clearly it is linear. Fix a set $K\subset {ℝ}^n\times ℝ$ and let $\phi \in C_0^{\infty }(K)$.

(c)

Why must there be a constant $T>0$ with

$$H(\phi )={\int \; <!--\; nolimits\; -->}_0^T{\int \; <!--\; nolimits\; -->}_{{ℝ}^n}\mathrm{\Phi }(x,t)\phi (x,t)\mathit{𝑑𝑥}\mathit{𝑑𝑡}?$$

(1 point)

(d)

Conclude with the help of Lemma 4.6 and Theorem 4.7 that

$$|H(\phi )|\le T\parallel \phi {\parallel }_{K,0}.$$

Hence $H$ is a continuous linear functional. (2 points)

Finally, we want to show that (in the sense of distributions) $({\partial }_t-\mathrm{\Delta })H=\delta$.

(e)

Extend Theorem 4.7 to show that

$${\int \; <!--\; nolimits\; -->}_{{ℝ}^n}\mathrm{\Phi }(x-y,t)h(y,s)\mathit{𝑑𝑦}\to h(x,s)$$

as $t\to 0$, uniformly in $s$. (1 point)

(f)

Hence show that

$${\int \; <!--\; nolimits\; -->}_{𝜀}^{\infty }{\int \; <!--\; nolimits\; -->}_{{ℝ}^n}\mathrm{\Phi }(-{\partial }_t\phi -\mathrm{\Delta }\phi )\mathit{𝑑𝑦}\mathit{𝑑𝑡}\to \phi (0,0)$$

as $𝜀\to 0$. (3 points)

(g)

Prove that as $𝜀\to 0$

$${\int \; <!--\; nolimits\; -->}_0^{𝜀}{\int \; <!--\; nolimits\; -->}_{{ℝ}^n}\mathrm{\Phi }(y,t)h(y,t)\mathit{𝑑𝑦}\mathit{𝑑𝑡}\to 0$$

(2 points)

Together these integrals show that

$$({\partial }_t-\mathrm{\Delta })H(\phi )=\left({\int \; <!--\; nolimits\; -->}_0^{𝜀}+{\int \; <!--\; nolimits\; -->}_{𝜀}^{\infty }\right){\int \; <!--\; nolimits\; -->}_{{ℝ}^n}\mathrm{\Phi }(-{\partial }_t\phi -\mathrm{\Delta }\phi )\mathit{𝑑𝑦}\mathit{𝑑𝑡}=\phi (0,0)=\delta (\phi )$$

for all test functions $\phi$. Therefore $({\partial }_t-\mathrm{\Delta })H=\delta$ as claimed.

34.

Heat death of the universe

First a corollary to Theorem 4.7:

(a)

Suppose that $h\in C_b({ℝ}^n)\cap L^1({ℝ}^n)$ and $u$ is defined as in Theorem 4.7. Show

$$\underset{x\in {ℝ}^n}{\sup <!--\; FUNCTION\; APPLICATION\; -->}|u(x,t)|\le \frac{1}{{(4\mathit{𝜋𝑡})}^{n∕2}}\parallel h{\parallel }_{L^1}.$$

(2 points)

The above corollary shows how solutions to the heat equation on ${ℝ}^n\times {ℝ}^{+}$ with such initial conditions behave: they tend to zero as $t\to \infty$. Physically this is because if $h\in L^1$ then there is a finite amount of total heat, which over time becomes evenly spread across the space.

On open and bounded domains $\mathrm{\Omega }\subset {ℝ}^n$ we can have different behaviour, due to the boundary conditions holding the temperature steady. In this question we determine the long time behaviour of solutions $u$ to the heat equation on open and bounded sets $\mathrm{\Omega }$ with $u(x,t)=0$ on $\partial \mathrm{\Omega }\times {ℝ}^{+}$ and $u(x,0)=h(x)$.We claim $u\to 0$ as $t\to \infty$.

(b)

Let $l_m$ be the function from Theorem 4.7 that solves heat equation on ${ℝ}^n$ with $l_m(x,0)=\mathit{𝑚𝑘}(x)$ for $m$ a constant and $k:{ℝ}^n\to [0,1]$ a smooth function of compact support such that $k{|}_{\mathrm{\Omega }}\equiv 1$. Why must $k$ exist? Why does $l_m\to 0$ as $t\to \infty$? What boundary conditions on $\mathrm{\Omega }$ does it obey? (3 points)

(c)

Use the monotonicity property to show that $u$ tends to zero. (2 points)

Hint. Consider $a=\underset{x\in \mathrm{\Omega }}{\sup <!--\; FUNCTION\; APPLICATION\; -->}|u(x,0)|$.