(You may see the rest of the lecture notes here).

Part I: Elliptic operators, local or not.

Today we wrap up the first part of the course, by proving the full existence theorem for weak solutions of (linear!) parabolic / kinetic equations generated by both differential and integro-differential elliptic operators.

Lax-Milgram and Lions-Magenes

In what follows we will work with a pair of Hilbert spaces, one denoted $\mathcal{H}$ and one denoted $\mathcal{E}$. In the examples that follow the spaces will be such that $\mathcal{E} \subset \mathcal{H}$ and continuously, that is, there is a constant $C>0$ such that

\begin{align} ||f||_{\mathcal{H}} \leq C ||f||_{\mathcal{E}}. \end{align}

Moreover, the set $\mathcal{E}$ will be a dense in $\mathcal{H}$ in the $\mathcal{H}$ norm. A key tool will be the following generalization of the Lax-Milgram theorem by Lions and Magenes.

Theorem. (Lions-Magenes)
Consider a Hilbert space $\mathcal{H}$, a Banach space $\mathcal{E}$, and a bilinear functional $\mathcal{B}:\mathcal{H}\times \mathcal{E} \to \mathbb{R}$ that is bounded, that is there is a $C_0>0$ such that \begin{align} B(f,g)\leq C_0||f||_{\mathcal{H}}||g||_{\mathcal{E}},\;\forall\;f\in \mathcal{H}, g\in\mathcal{E} \end{align} Then, the following two conditions are equivalent
1) There is a number $\alpha>0$ such that \begin{align} \sup_{||f||_{\mathcal{H}} \leq 1} |\mathcal{B}(f,g)| \geq \alpha||g||_{\mathcal{E}} \;\forall\;g\in\mathcal{E}. \end{align} 1) For any $\phi \in \mathcal{E}^*$ there is at least one $f\in\mathcal{H}$ such that \begin{align} \mathcal{B}(f,g) = \phi(g) \;\;\forall\;\phi \in \mathcal{E}. \end{align}

Proof.
Proof that 1) implies 2). We will use $\mathcal{L}:\mathcal{H} \mapsto \mathcal{E}^*$ to denote the natural map defined by $\mathcal{B}$, that is, given $f\in\mathcal{H}$ we have \begin{align} \mathcal{L}(f)(g) := \mathcal{B}(f,g),\;\forall\;g\in \mathcal{E}. \end{align} The adjoint to this map, $\mathcal{L}^*:\mathcal{E} \to \mathcal{H}^*$ is given by \begin{align} \mathcal{L}^*(g)(f) := \mathcal{B}(f,g),\;\forall\;f\in \mathcal{H}, \end{align} The boundedness of the bilinear form $\mathcal{B}$ amounts to the fact that $\mathcal{L}$ (and $\mathcal{L}^*$) are bounded, \begin{align} ||\mathcal{L}f||_{\mathcal{E}^*} \leq C_0||f||_{\mathcal{H}} \end{align} Now, the inequality in condition 1) is equivalent to the statement \begin{align} ||\mathcal{L}^*g||_{\mathcal{H}} \geq \alpha||g||_{\mathcal{E}} \end{align} It follows that $\mathcal{L}^*$ is injective and its image is a closed subspace of $\mathcal{H}$. From the latter fact follows that the range of $\mathcal{L}$ is all of $\mathcal{E}^*$, and this proves 2).

Proof that 2) implies 1). Suppose 1) fails, so that for any $k\in\mathbb{N}$ there is a $g_k \in \mathcal{E}$ such that \begin{align} \sup_{||f||_{\mathcal{H}}\leq 1} |\mathcal{B}(f,g_k)| \leq k^{-1}||g_k||_{\mathcal{E}} \end{align} By normalizing, we may assume without loss of generality that $||g_k||_{\mathcal{E}}=1$ for each $k$. \begin{align} \sup_{||f||_{\mathcal{H}}\leq 1} |\mathcal{B}(f,kg_k)| \leq 1 \end{align} From here, we argue by contradiction that 2) cannot hold, for if it did, the last inequality above would imply that for any $\phi \in \mathcal{E}^*$ we have \begin{align} \phi(kg_k) \leq C_\phi\;\forall\; k \in \mathbb{N}. \end{align} This implies the sequence $\{kg_k\}_k$ is weakly bounded, which is the same as it being bounded, a contradiction with the fact that $||g_k||_{\mathcal{E}}=1$ for all $k$.

A very general existence and uniqueness result

Now let us describe the setting where this theorem is applied.

First, we describe the functional spaces that are part of the description of the problem. First, one has a Hilbert space $H$ lying inside $L^2(\mathbb{R}^n)$, such that for some $C>0$ we have

\begin{align} ||f||_{L^2} \leq C||f||_{H}. \end{align}

Furthermore, we shall assume that $C^\infty_c(\mathbb{R}^n)$ is a dense subset of $H$ (note that, among other things, this guarantees $H$ is also a dense subset of $L^2(\mathbb{R}^n))$.

For a fixed $M>0$ we define the following spaces \begin{align} \mathcal{H} & = \left \{ f:[0,T]\times \mathbb{R}^n \to \mathbb{R} \;:\; \int_0^\infty e^{-Mt}||f(t)||_{H}^2dt <\infty \right \} \\ \
\mathcal{E} & = \left \{ f:[0,T]\times \mathbb{R}^n \to \mathbb{R} \;:\; \int_0^\infty e^{-Mt} (||f(t)||^2_{H}+ ||\partial_tf(t)||_{H^*}^2 ) dt < \infty \right \} \end{align}

These come with the respective Hilbert space norms defined by

\begin{align} ||f||_{\mathcal{H}}^2 & = \int_0^\infty e^{-Mt}||f(t)||^2_{H}dt\\ \
||f||^2_{\mathcal{E}} & = \int_0^\infty e^{-Mt} (||f(t)||_{H}^2+||\partial_tf(t)||^2_{H^*})dt \end{align}

Now we describe the structure that will describe the underlying PDE. Suppose we are given a family of bilinear functionals $B_t: H \times H \to \mathbb{R}$ where $t>0$ which satisfies the following three conditions

1. Given $f\in \mathcal{H}$ and $g\in\mathcal{E}$ the function $t\mapsto B_t(f(t),g(t))$ is measurable.
2. There is constant $C_0>0$ such that \begin{align} |B_t(f,g)| \leq C_0||f||_{H}||g||_{H} \;\text{ for } a.e.\;t \end{align}
3. There are constants $c_0,C_1>0$ such that for any $g\in H$ we have \begin{align}
   B_t(g,g) \geq c_0||g||_{H}^2 -C_1||g||_{L^2}^2\;\text{ for } a.e.\;t \end{align}

Theorem. (Existence)
Consider a family of bilinear functionals $B_t$ satisfying the three conditions above, and choose $m=2C_1$. Then given any $f_{\text{in}} \in L^2(\mathbb{R}^n)$ there is a function $f:(0,\infty)\times \mathbb{R}^n\to\mathbb{R}$ with $f \in \mathcal{H}$ such that \begin{align} \int_0^\infty \int_{\mathbb{R}^n} f(-\partial_t \phi) + B_t(f,\phi)\;dxdt = \int_{\mathbb{R}^n}f_{\text{in}}(x)\phi(x,0)\;dx, \end{align} for all functions $\phi \in C^\infty_c([0,\infty)\times \mathbb{R}^n)$.

Proof (part 1 of 3). We define the bilinear form \begin{align} \mathcal{B}(f,g) = \int_0^\infty (f(t),-\partial_t(e^{-Mt}g(t))) + B_t(f(t),e^{-Mt}g(t))dt \end{align} We are going to show there is a unique $f\in \mathcal{H}_1$ such that \begin{align} \mathcal{B}(f,g) = \int_{\mathbb{R}^n}f_{\text{in}}(x)g(x,0)\;dx \;\forall\;g\in \mathcal{E} \end{align} This would prove the theorem, as in this case given $\phi \in C^\infty_c([0,\infty)\times \mathbb{R}^n)$ we can take $g = e^{Mt}\phi$ which is in $\mathcal{E}$, and so \begin{align} \int_0^\infty (f(t),-\partial_t\phi(t)) + B_t(f(t),\phi(t))dt = \int_{\mathbb{R}^n}f_{\text{in}}(x)\phi(x,0)\;dx \;\forall\;g\in \mathcal{E}. \end{align}

Proof (part 2 of 4). We now prove $\mathcal{B}$ satisfies the boundedness condition. Observe that \begin{align} \langle f(t),\partial_t(e^{-Mt}g(t))\rangle = e^{-Mt}\langle f(t),\partial_tg(t) \rangle - M e^{-Mt}\langle f(t),g(t)\rangle, \end{align} and so $\mathcal{B}$ can be written as the sum of three terms, \begin{align} \mathcal{B}(f,g) & = -\int_0^\infty e^{-Mt}\langle f(t),\partial_tg(t)\rangle \;dt + M\int_0^\infty e^{-Mt}\langle f(t),g(t)\rangle dt \\\ \\\ & \;\;\;\;+ \int_0^\infty e^{-Mt}B_t(f(t),g(t))dt. \end{align} Now we estimate each term one by one \begin{align} \left |\int_0^\infty e^{-Mt}\langle f(t),\partial_t g(t)\rangle dt\right | & \leq \int_0^\infty e^{-Mt}||f(t)||_{H}||\partial_t g(t)||_{H^*}dt \\\ \\\ & \leq ||f||_{\mathcal{H}} ||g||_{\mathcal{E}},\\\ \\\ \left |\int_0^\infty e^{-Mt}\langle f(t),g(t)\rangle \;dt \right | & \leq \int_0^\infty e^{-Mt}||f(t)||_{H} ||g(t)||_{H}\;dt \\\ \\\ & \leq C ||f||_{\mathcal{H}_1}||g||_{\mathcal{E}},\\\ \\\ \left | \int_0^\infty e^{-Mt}B_t(f,g)dt \right | & \leq C_0 \int_0^\infty e^{-Mt} ||f(t)||_{H} ||g(t)||_{H}dt \\\ \\\ & \leq C_0 ||f||_{\mathcal{H}} ||g||_{\mathcal{E}}, \end{align} where in each three instances we used the Cauchy-Schwartz inequality. Putting these inequality together we have \begin{align} |\mathcal{B}(f,g)| \leq (1+CM+C_0) ||f||_{\mathcal{H}}||g||_{\mathcal{E}}, \end{align} and so $\mathcal{B}$ is a bounded bilinear form.

For the last two steps in the proof, we will need a Lemma about the derivative in time of the $L^2$ norm of a function in $\mathcal{E}$, this is due to Lions and Magenes (who stated it for $M=0$).

The lemma uses the fact that $H\subset L^2$ implies that $L^2 \subset H^*$, and so if we have a function $f\in L^2((0,T),H)$ such that in addition $\partial_t f \in L^2((0,T),H^*)$, we can then make sense for a.e. $t$ of the expression

\begin{align} \langle f(t),\partial_tf(t)\rangle, \end{align} which defines an integrable function of $t$ in $(0,T)$. Then, using that $H \subset L^2 \subset H^*$ one shows that

\begin{align} \frac{d}{dt}||f(t)||^2_{L^2} = 2\langle f(t),\partial_tf(t)\rangle. \end{align}

We state the lemma and omit the proof for now.

Lemma. (Lions-Magenes) For any $g \in \mathcal{H}$ we have \begin{align} -\int_0^\infty e^{-Mt}\langle g(t),\partial_t g(t)\rangle dt = \frac{1}{2}||g(0)||^2_{2} - \frac{M}{2}\int_0^\infty e^{-Mt}||g(t)||^2_2\;dt. \end{align}

With this identity in hand, we return to the proof of the theorem.

Proof (part 3 of 4) . It remains to show that there exists a number $\alpha>0$ such that \begin{align} \sup \limits_{||f||_{\mathcal{H}} \leq 1}\mathcal{B}(f,g) \geq \alpha \;\text{ for any } g \text{ s.t. } ||g||_{\mathcal{E}}=1. \end{align} Suppose no such $\alpha$ exists, then for any $k\in\mathbb{N}$ there is a $g_k\in\mathcal{E}$ such that $||g_k||_{\mathcal{E}} = 1$ and \begin{align} \sup \limits_{||f||_{\mathcal{H}}=1} \mathcal{B}(f,g_k) \leq \frac{1}{k} \end{align} First, we claim that for any $g \in \mathcal{E}$, \begin{align} \mathcal{B}(g,g) \geq c_0\int_0^\infty e^{-Mt}||g(t)||_{H}^2\;dt = c_0 ||g||_{\mathcal{H}}^2. \end{align} Indeed, note that \begin{align} \mathcal{B}(g,g) = \frac{1}{2}||g(0)||_2^2 + \int_0^\infty e^{-Mt}\left ( \frac{1}{2}M||g(t)||_{L^2}^2+B_t(g(t),g(t)) \right )\;dt, \end{align} and then, since $M = 2C_1$, \begin{align} \mathcal{B}(g,g) \geq c_0\int_0^\infty e^{-Mt}||g(t)||_{H}^2\;dt, \end{align} as claimed. Applying this to the sequence $g_k$ we conclude that \begin{align} ||g_k ||_{\mathcal{H}} \to 0 \text{ as } k \to \infty \end{align} On the other hand, for general $f,g$ we have \begin{align} \mathcal{B}(f,g) \geq \int_0^\infty e^{-Mt}(f(t),-\partial_tg(t))\;dt - C_0\int_0^\infty e^{-Mt}||f(t)||_H ||g(t)||_H\;dt \end{align} In particular, whenever $||f||_{\mathcal{H}} \leq 1$ and $g = g_k$ this implies that \begin{align} \mathcal{B}(f,g_k) \geq \int_0^\infty e^{-Mt}(f(t),-\partial_tg_k(t))\;dt - C_0||g_k||_{\mathcal{H}} \end{align} Then, given that $||g_k||_{\mathcal{H}}$ is converging to zero, \begin{align} \lim \limits_{k\to \infty} \sup \limits_{||f||_{\mathcal{H}}=1}\mathcal{B}(f,g_k) \geq \lim \limits_{k\to \infty} \sup \limits_{||f||_{\mathcal{H}}=1}\mathcal{B}(f,g_k) \int_0^\infty e^{-Mt}(f(t),-\partial_tg_k(t))\;dt \end{align} On the other hand, by duality \begin{align} \sup\limits_{||f||_{\mathcal{H}}\leq 1}\int_0^\infty e^{-Mt}(f(t),-\partial_tg_k(t))\;dt = \left (\int_0^\infty e^{-Mt}||\partial_t g_k||_{H^*}^2dt\right )^{1/2} \end{align} Since $||g_k||_{\mathcal{E}}=1$ and $||g_k||_{\mathcal{H}} \to 0$, it follows that \begin{align} \int_0^\infty e^{-Mt}||\partial_t g_k||_{H^*}^2dt \to 1 \text{ as } k \to \infty. \end{align} We conclude that \begin{align} \lim \limits_{k\to \infty} \sup \limits_{||f||_{\mathcal{H}}\leq 1}\mathcal{B}(f,g_k) \geq 1, \end{align} which is a contradiction.

Proof (part 4 of 4) . It remains to prove that \begin{align} g \mapsto \int_{\mathbb{R}^n}g(x,0)f_{\text{in}}(x)dx \end{align} is a bounded linear functional in $\mathcal{E}$. Here is a point where the extra regularity of functions belonging to the class $\mathcal{E}$ comes into play, for, as we saw earlier for the heat equation, having $\partial_tg \in L^2_{\text{loc}}(H)^*$ makes $t\mapsto g(t)$ a continuous map from $[0,\infty)$ to $(H)^*$. In particular, we have seen that if we fix $\phi \in H$, then \begin{align} |\langle g(t_2,\cdot),\phi\rangle - \langle g(t_1,\cdot),\phi\rangle | \leq |t_2-t_1|^{1/2}||g||_{\mathcal{E}}||\phi||_{H},\;\forall\;t_1,t_2\geq 0. \end{align} In particular, for any $t>0$ we have \begin{align} \langle g(0,\cdot),\phi\rangle \leq |t|^{1/2}||g||_{\mathcal{E}}||\phi||_{H}+\langle g(t,\cdot),\phi\rangle \end{align} Taking an average over $t\in(0,1)$, we have \begin{align} \langle g(0,\cdot),\phi\rangle & \leq ||g||_{\mathcal{E}}||\phi||_{H}+\int_0^1\langle g(t,\cdot),\phi\rangle\;dt \\\ \\\ & \leq 2||g||_{\mathcal{E}}||\phi||_{H}. \end{align} This shows that for $\phi \in \mathcal{H}$, the functional $g\mapsto \langle g(0,\cdot),\phi \rangle$ is a bounded linear functional in $\mathcal{E}$. From here follows the existence of a solution when we have initial data $f_{\text{in}} \in H$. The general case $f \in L^2(\mathbb{R}^n)$ follows by a density argument, making use of the fact that once we have built a solution with initial data in $H$, the solution has a bound in the $\mathcal{H}$ norm that depends only on the $L^2$ norm of the initial data. We skip the details.

We have now established the existence of weak solutions to our abstract evolution problem, what about uniqueness? The linearity of the equation makes this much simpler than it is general.

If $f_1$ and $f_2$ are two solutions to the problem with initial data $f_{\text{in},1}$ and $f_{\text{in},2}$, respectively, then $\alpha_1f_1+\alpha_2f_2$ will be a weak solution with initial data $\alpha_1 f_{\text{in},1}+\alpha_2 f_{\text{in},2}$. In particular, if $f_1$ and $f_2$ are two weak solutions with the same initial data then $f_1-f_2$ will be a weak solution whose initial data is zero everywhere.

Theorem. (Uniqueness)
Assume $H_1$ and $H_2$ are the same space, and take $B_t$ satisfying the three conditions above. We also fix some $f_{\text{in}} \in L^2(\mathbb{R}^n)$.
Fix $m$ equal to $2C_1$, suppose there are two functions $f_1,f_2:(0,\infty)\times \mathbb{R}^n\to\mathbb{R}$ with $f_1,f_2 \in \mathcal{H}$ such that \begin{align} \int_0^\infty \int_{\mathbb{R}^n} f_i(-\partial_t \phi) + B_t(f_i,\phi)\;dxdt = \int_{\mathbb{R}^n}f_{\text{in}}(x)\phi(x,0)\;dx, \text{ for } i=1,2, \end{align} for all functions $\phi \in C^\infty_c([0,\infty)\times \mathbb{R}^n)$. Then $f_1=f_2$ a.e. in $(0,\infty)\times \mathbb{R}^n$.

Sketch of the proof. The function $w = f_1-f_2$ is a weak solution corresponding to zero initial data. The energy inequality says \begin{align} 0 = \frac{1}{2}||w(0)||_2^2 \geq \mathcal{B}(w,w) \geq \lambda \int_0^\infty e^{-Mt}||w(t)||_{H}^2\;dt \end{align} This shows that $||w(t)||_{H}=0$ for a.e. $t$, and so $f_1=f_2$ for a.e. $(t,x)\in(0,\infty)\times \mathbb{R}^n$.

With the existence and uniqueness theorem in hand, we are going to see what it looks like when applied to second order (local) parabolic equations and integro-differential (non-local) parabolic equations.

What this theorem says about local parabolic equations

To illustrate the power of this theorem, let us solve the Cauchy problem for \begin{align} \partial_tf & = \text{div}(A\nabla f+fb) \\ \
f(x,0) & = f_{\text{in}} \end{align} for a general $f_{\text{in}} \in L^2(\mathbb{R}^n)$ and where $A = A(t,x)$ and $b = b(t,x)$. The only assumptions we make on the coefficients is that they be measurable, and that there exist $0<\lambda \leq \Lambda$ such that \begin{align} \lambda \text{I} \leq A(t,x) \leq \Lambda \text{I},\; |\text{div}(b)| \leq \Lambda \;\forall\; (t,x). \end{align}

We now check the underlying bilinear form satisfies the assumptions of the theorem. Given $f,g\in H^1(\mathbb{R}^n)$ and we define for a.e. $t>0$

\begin{align} B_t(f,g) = \int_{\mathbb{R}^n}(A(t,x)\nabla f+fb(t,x),\nabla g) dx \end{align}

Observe that \begin{align} B_t(g,g) & = \int_{\mathbb{R}^n}(A\nabla g,\nabla g)\;dx +\frac{1}{2}\int_{\mathbb{R}^n} (b,\nabla g^2)\;dx \\ \
& = \int_{\mathbb{R}^n}(A\nabla g,\nabla g)\;dx -\frac{1}{2}\int_{\mathbb{R}^n} \text{div}(b) g^2\;dx \end{align}

Therefore, with $C_1 = \tfrac{1}{2}||\text{div}(b)||_\infty$ we have

\begin{align} B_t(g,g) & \geq \lambda \int_{\mathbb{R}^n}|\nabla g|^2\;dx -C_1 \int_{\mathbb{R}^n}g^2\;dx = \lambda ||g||_{H^1}^2 -(C_1+\lambda) ||g(t)||_2^2 \end{align}

On the other hand

\begin{align} \left | \int_{\mathbb{R}^n}(A\nabla f,\nabla \phi) dx \right | & \leq \left ( \int_{\mathbb{R}^n}(A\nabla f,\nabla f)\;dx\right )^{1/2}\left ( \int_{\mathbb{R}^n}(A\nabla \phi,\nabla \phi)\;dx\right )^{1/2} \\ \
& \leq \Lambda \left ( \int_{\mathbb{R}^n}|\nabla f|^2\;dx\right )^{1/2}\left ( \int_{\mathbb{R}^n}|\nabla \phi|^2\;dx\right )^{1/2} \\ \
& \leq \Lambda ||f||_{H^1} ||g||_{H^1} \end{align}

Then the assumptions of the theorems hold, and we have the following

Theorem. For any $f_{\text{in}} \in L^2(\mathbb{R}^n)$ there is a unique function $f(t,x)$ satisfying the integral bound \begin{align} \int_0^\infty e^{-Mt}||f(t)||^2_{H^1}\;dt < \infty, \;M=2(C_1+\lambda), \end{align} and such that for any $\phi \in C^\infty_c([0,\infty)\times \mathbb{R}^n)$ we have \begin{align} \int_0^\infty\int_{\mathbb{R}^n}f(-\partial_t\phi) + (A\nabla f+f b,\nabla \phi)dxdt = \int_{\mathbb{R}^n}f_{\text{in}}(x)\phi(x,0)\;dx. \end{align}

Remark. A posteriori, one can check that the unique function $f$ provided by the lemma is automatically a bit regular in time with respect to the $(H^1)^*$ norm, namely we have that $\partial_t f \in L^2_{\text{loc}}(H^1)^*$.

What this theorem says about integro-differential parabolic equations

Define the form \begin{align} B_t(f,g) & = \frac{1}{2}\iint_{\mathbb{R}^{2n}}(f(x)-f(y))(g(x)-g(y))K_t(x,y) dxdy + \int_{\mathbb{R}^n}U(t,x)fg\;dx,
\end{align} with $U \in L^\infty$, and \begin{align} \frac{\lambda c_{n,s}}{|x-y|^{n+2s}}\leq K_t(x,y) \leq \frac{\Lambda c_{n,s}}{|x-y|^{n+2s}}\text{ for a.e. } t,x,y, \end{align} for some constants $0<\lambda\leq \Lambda$.

As before, we now check the time-dependent bilinear form $B_t$ satisfies the assumptions of the existence theorem.

\begin{align} B_t(g,g) & = \frac{1}{2}\iint_{\mathbb{R}^{2n}}(g(x)-g(y))^2K_t(x,y)dxdy + \int_{\mathbb{R}^n} U(t,x)g(x)^2dx \end{align}

From the assumption on the kernel, we have \begin{align} \frac{1}{2}\iint_{\mathbb{R}^{2n}}(g(x)-g(y))^2K_t(x,y)dxdy \geq \lambda \frac{c_{n,s}}{2}\iint_{\mathbb{R}^{2n}}\frac{(g(x)-g(y))^2}{|x-y|^{n+2s}}dxdy = \lambda ||g||_{\dot H^s}^2 \end{align}

On the other hand, using that $U$ is in $L^\infty((0,\infty)\times \mathbb{R}^n)$, \begin{align} \int_{\mathbb{R}^n} U(t,x)g(x)^2dx \geq -||U||_{\infty} ||g||_2^2 \end{align}

These inequalities amount to

\begin{align} B_t(g,g) & \geq \lambda ||g||^2_{\dot H^s} - ||U||_{\infty}||g||^2_2 \\ \
& = \lambda ||g||^2_{H^s} -(\lambda+||U||_{\infty})||g||^2_2. \end{align}

Theorem. There is a unique function $f(t,x)$ satisfying the bound \begin{align} \int_0^\infty e^{-Mt}||f(t)||^2_{H^s}dt < \infty,\; M = 2(||U||_\infty+\lambda), \end{align} and such that for any $\phi \in C^\infty_c([0,\infty)\times \mathbb{R}^n)$ we have \begin{align} & \int_0^\infty\int_{\mathbb{R}^n}f(-\partial_t\phi)dxdt + \int_0^\infty B_t(f(t),\phi(t)) dt = \int_{\mathbb{R}^n}f_{\text{in}}(x)\phi(x,0)\;dx. \end{align}

Remark. Just as in the local, second order case, one can check this weak solution automatically enjoys some weak regularity in time, in this case $\partial_t f \in L^2_{\text{loc}}(H^s)^*$.

Bonus: Kinetic equations

We will not go over the details right now, but it is worth exploring how the above approach based on the Lions-Magenes theorem can be used for linear kinetic equations.

That is we are looking at functions $f=f(t,x,v)$ solving

\begin{align} \partial_t f + v\cdot \nabla_x f = Lf \end{align} where $Lf = L(f(t,x,\cdot))$ is any of the linear elliptic operators we have covered in the parabolic theory, e.g. $(L f) (t,x,v)= \Delta_v f(t,x,v)$.

Since the elliptic operator acts only on the $v$ variable, the pertinent space $H$ for this case should be something of the type \begin{align} H^1_v(\mathbb{R}^{2n}) = \left \{ f:\mathbb{R}^{2n} \to \mathbb{R} \;:\; f(x,v) \in L^2(\mathbb{R}^{2n}) \text{ and } \nabla_v f(x,v) \in L^2(\mathbb{R}^{2n}) \right \} \end{align} with respective norm, \begin{align} ||f||_{H^1_v}^2 = \iint_{\mathbb{R}^{2n}}f(x,v)^2\;dxdv + \iint_{\mathbb{R}^{2n}}|\nabla_v f(x,v)|^2\;dxdv \end{align} More generally, for $s\in(0,1]$, we define

\begin{align} H^s_v(\mathbb{R}^{2n}) = \left \{ f:\mathbb{R}^{2n} \to \mathbb{R} \;:\; f(x,v\cdot) \in H^s(\mathbb{R}^{n}) \text{ for } a.e. x \text{ and } \int_{\mathbb{R}^n}||f(x,\cdot)||_{H^s}^2\;dx < \infty \right \} \end{align} which would be a relevant space for many integro-differential operators, and its respective norm is

\begin{align} ||f||_{H^s_v}^2 = \iint_{\mathbb{R}^{2n}}f(x,v)^2\;dxdv + \int_{\mathbb{R}^n}\left ( \iint_{\mathbb{R}^{2n}}\frac{(f(x,v)-f(v,w))^2}{|v-w|^{n+2s}}\;dvdw\right ) dx. \end{align}

Having established these spaces, for $M>0$ we would now define the spaces $\mathcal{H}$ and $\mathcal{E}$ where our bilinear form $\mathcal{B}$ will be defined. The appearance of the term $v\cdot \nabla_x $ in the equation means we must contend with it somewhere in one of the spaces, after some reflection one sees that one must treat $\partial_t$ and $v\cdot \nabla_x$ together. In fact, it will be convenient to introduce some notation and define the operator acting on functions $\phi(t,x,v)$ as follows \begin{align} D_{t,v}\phi(t,x,v) = \partial_t \phi(t,x,v) + v\cdot \nabla_x \phi(t,x,v) \end{align}

With this in hand, we define the spaces \begin{align} \mathcal{H} & = \left \{ f:(0,\infty)\times \mathbb{R}^{2n}\to\mathbb{R} \;:\; \int_0^\infty e^{-Mt}||f(t)||^2_{H^s_v}dt < \infty \right \} \\ \
\mathcal{E} & = \left \{ f\in \mathcal{H} \;:\; \int_0^\infty e^{-Mt}\left ( ||D_{t,v} f||^2_{(H^s_v)^*}\right ) dt < \infty \right \}, \end{align} and the bilinear form $\mathcal{B}:\mathcal{H}\times \mathcal{E} \to \mathbb{R}$ \begin{align} \mathcal{B}(f,g) = \int_0^{\infty} \langle f,-D_{t,v}(e^{-Mt}g)\rangle +e^{-Mt}B_t(f(t),g(t))\;dt \end{align} where $B_t$ is one of the time-dependent bilinear forms we have worked with above.

This post is already too long, so we will figure out the rest in a future lecture (probably right after we are done with Part II and start the kinetic regularity theory section). Things left to check include checking the Lions-Magenes theorem applies to $\mathcal{B}$ and the spaces above, and that the trace at $t=0$ of a function in $\mathcal{E}$ is well defined (so that an initial data in $L^2$ can be used to define a bounded linear functional), this latter point might require using identities of the form

\begin{align} \frac{d}{ds}\iint_{\mathbb{R}^{2n}}g(t,x,v)\phi(x-tv,v)dxdv = \langle D_{t,v}g,\phi(\cdot-t\cdot,\cdot)\rangle \end{align}

where $\phi$ is a smooth test function.