On Kolmogorov Fokker Planck operators with linear drift and time dependent measurable coefficients

1.   Introduction and main results

Starting from the theory present in ^[2,3,4], we shall prove the well-posedness of a global Cauchy problem for a class of Kolmogorov-Fokker-Planck operators (briefly KFP) with time dependent measurable coefficients and linear drift (Theorem 1.1). The main point is to prove the existence of a solution because uniqueness and the stability estimates follow from ^[2,3]. Moreover, since we have at our disposal an explicit fundamental solution (see ^[4]) we shall look for a solution in the form prescribed by the Duhamel method. To this aim, we first employ some techniques from ^[2] to obtain existence for smooth datum and then thanks to the estimates from ^[3], we can achieve existence under the minimal regularity assumptions on the datum.

The KFP operator we consider is defined as follows:

for some $q\leq N$. Moreover we assume the two following hypotheses:

$h1)$ The coefficients $a_{ij}$ are measurable functions and there exists $\nu > 0$ such that the matrix $\mathbb{A}(t) = \{a_{ij}(t)\}_{i, j = 1}^q$ satisfies the following condition:

$h2)$ There exist positive integers $\{m_j\}_{j = 1}^\kappa$ satisfying $q = m_0\geq m_1\geq\dots\geq m_\kappa$ and $\sum_{j = 1}^\kappa m_j = N$ such that the matrix $\mathbb{B} = \{b_{ik}\}_{i, k = 1}^N$ assumes the following block structure:

where for every $j\in \{1, \dots, \kappa\}$ the block $\mathbb{B}_j$ has dimension $m_j\times m_{j-1}$ and rank equal to $m_j$.

Remark 1.1. It is convenient to introduce the first order operator

so that (1.1) becomes: $\mathcal{L} = \sum_{i, j = 1}^q a_{ij}(t)\partial_{x_ix_j}^2+Y$.

As pointed out by Bramanti and Polidoro in ^[4] this class of operators is naturally linked to stochastic systems of the kind:

Indeed, taking $a_{ij}(t) = \frac{1}{2}\sum_{k = 1}^q\sigma_{ik}(t)\sigma_{jk}(t)$, the forward Kolmogorov operator of this system corresponds to $\mathcal{L}$ while the backward Kolmogorov operator corresponds to the adjoint of $\mathcal{L}$. In accordance to this possible application we remark that the simple case

has been studied, already in 1934, by Kolmogorov in relation to a system with $2N$ degrees of freedom ^[6] (see [5, Section 2]). It is interesting to notice that this operator is hypoelliptic and admits an explicit fundamental solution found, by Kolmogorov himself, which is smooth outside the pole. The operator (1.4) is a particular case of the one studied by Lanconelli and Polidoro in the fundamental paper ^[8] which contains a characterization of the hypoellipticity for the more general operator

where $\{\tilde{a}_{ij}\}_{i, j = 1}^N$ and $\{\tilde{b}_{ij}\}_{i, j = 1}^N$ are constant matrices. Actually (see ^[8]), the operator (1.5) is hypoelliptic if and only if there exists a change of variables which leads to an operator of the kind (1.1) whose matrix $\mathbb{A}$ is constant and positive definite while $\mathbb{B}$ has a structure similar to (1.3) but with blocks above the $\mathbb{B}_j$ ones arbitrarily chosen:

Moreover, the mentioned paper points out the following homogeneous group structure which is fundamental for the study of this kind of operators.

Definition 1.1. Let $E(s): = e^{-s\mathbb{B}}$ and let

The homogeneous group structure we consider is given by the group law $\circ$

and the family of automorphisms $\{D(\lambda)\}_{\lambda > 0}$

From this definition it easily follows that $(\mathbb{R}^{N+1}, \circ)$ is a Lie group:

and under the assumption (h2) (see [8, Remark 2.1]) for any $(x, t)\in \mathbb{R}^{N+1}$ and $(y, s)\in \mathbb{R}^{N+1}$ we have

Moreover, the homogeneous group defined above admits a metric structure.

Definition 1.2. We define the homogeneous norm $\rho:\mathbb{R}^{N+1}\to[0, +\infty)$ as

Thanks to $\rho$ we can define a quasi distance $d$ on $\mathbb{R}^{N+1}$ as follows:

From this definition it immediately follows that for any $(x, t)\in \mathbb{R}^{N+1}$ and $\lambda\in(0, +\infty)$ we have:

while for $\xi$, $\zeta$ and $\eta$ in $\mathbb{R}^{N+1}$ we have:

Now we recall some definitions that can be found in ^[3].

Definition 1.3. Let $I$ be an interval and let $\Omega = \mathbb{R}^N\times I$.

For any measurable $f:\Omega\to\mathbb{R}$ we define:

Then the function $f$ is said partially Dini continuous if

Finally, the space of functions $f\in L^\infty(\Omega)$ which are partially Dini continuous is denoted with $\mathcal{D}(\Omega)$.

Definition 1.4. Let $I$ and $\Omega$ be as in the previous definition and let $\mu > 0$. For $f\in \mathcal{D}(\Omega)$ let $\mathcal{M}_{f, \Omega}$ and $\mathcal{U}^\mu_{f, \Omega}$ be defined as follows:

Remark 1.2. Concerning $\mathcal{M}_{f, \Omega}(r)$ and $\mathcal{U}_{f, \Omega}^\mu(r)$ we remark only that for $f\in \mathcal{D}(\Omega)$ they are two modului of continuity (i.e., monotone nondecreasing functions vanishing as $r\to 0^+$). Moreover whenever $f$ is log-Dini continuous, meaning that its modulus of continuity satisfies:

the function $\mathcal{M}_{f, \Omega}$ is a Dini continuity modulus. See [3, Section 2] for details concerning these definitions and in particular for the assertions above, see Proposition 2.8 and Lemma 2.12.

Let us introduce the following definitions concerning the spaces in which we look for a solution.

Definition 1.5. Let $I$ be an open interval, then we define the spaces

and

The partial derivatives in the definition of $S^0(\mathbb{R}^N\times I)$ are distributional derivatives.

For simplicity we adopt the notation

and

for $\tau < T$ real numbers. Now we can state the main result of this paper.

Theorem 1.1 (Well-posedness of the Cauchy problem). Let $f\in \mathcal{D}(S_{+\infty})$ with ${\rm{supp}}(f)\subset \mathbb{R}^N\times [0, +\infty)$ and let $g\in \mathcal{D}(\mathbb{R}^N)$ satisfy $\partial_{x_i x_j}g\in \mathcal{D}(\mathbb{R}^N)$ for $i, j\in\{1, \dots, q\}$ and $Yg\in\mathcal{D}(\mathbb{R}^N)$ (the derivatives are taken in the distributional sense). Then, there exists a unique $u\in \mathbb{S}(0, +\infty)$ solution of the Cauchy problem:

Moreover, the solution $u$ is in the form:

and in addition there exist two constants $c$ and $\mu$ depending only on $\nu$ and $\mathbb{B}$ such that for any $T > 0$ the following estimates hold:

Remark 1.3. Concerning the existence of a solution the assumptions on the datum $g$ could be significantly weakened. Indeed, the existence for the homogeneous problem has been studied under very weak assumptions by Bramanti and Polidoro in ^[4].

2.   Preliminaries and known results

This section recalls some known results about the operator (1.1) first assuming the matrix $\mathbb{A} = \{a_{ij}\}_{i, j = 1}^q$ constant and then for the more general case. The main references are the papers ^[2,3,4,7,8].

If $\mathbb{A}$ is constant, the operator (1.1) enjoys nice properties related to the homogeneous group, which remind the ones of the heat operator (see ^[7,8]). We shall list some of them in the next theorem. We also define the homogeneous dimension:

where the coefficients $\{m_j\}_{j = 1}^\kappa$ are defined in (1.3).

Theorem 2.1 (^[8]). Assumes $(h1)$ and $(h2)$. If the matrix $\mathbb{A} = \{a_{ij}\}_{i, j = 1}^q$ is constant the operator (1.1) satisfies the following properties:

i) $\mathcal{L}$ is hypoelliptic;

ii) $\mathcal{L}$ is invariant with respect to left translations in $(\mathbb{R}^{n+1}, \circ)$: Let $(y, s)\in \mathbb{R}^{N+1}$ and $u\in C_0^\infty(\mathbb{R}^{N+1})$, then, for any $(x, t)\in\mathbb{R}^{N+1}$

iii) $\mathcal{L}$ is $D(\lambda)$-homogeneous of degree 2: for every $\lambda > 0$ and $u\in C_0^\infty(\mathbb{R}^{N+1})$

iv) Let $C(1)$ and $c(1)$ be defined as:

and let

be defined by

Then, for any fixed $(y, s)\in \mathbb{R}^{N+1}$ the function $\Gamma(\cdot; y, s)$ belongs to $C^\infty(\mathbb{R}^{N+1}\setminus\{(y, s)\})$ and is the fundamental solution with pole $(y, s)$:

and for any $g\in C_0(\mathbb{R}^N)$:

v) For any $(y, s), (x, t)\in \mathbb{R}^{N+1}$ if $t > s$ we have:

vi) For any $(x, t), (y, s)\in \mathbb{R}^{N+1}$ such that $(x, t)\not = (y, s)$

vii) For any $\lambda > 0$ and $(x, t)\in \mathbb{R}^{N+1}\setminus\{(0, 0)\}$

When $\mathbb{A}$ is a multiple of the identity the fundamental solution just introduced enters many computations, hence, we introduce the following notation.

Notation 2.1. Let $a > 0$, $\Gamma_a$ shall denote the fundamental solution of

Moreover, we define

and

With this notation, for $(x, t), (y, s)\in \mathbb{R}^{N+1}$ with $t > s$, we have:

Now we turn our attention to the case of varying coefficients. The problem of finding a fundamental solution for operators with nonconstant matrix $\mathbb{A} = \{a_{ij}\}_{i, j = 1}^q$ has been studied by various authors, see ^{[4,5,7,9,10,11]}. However among the mentioned papers only the last two assume coefficients depending on time in a nonsmooth way, here we recall some results from ^[4].

Theorem 2.2. [4, Theorem 1.4] Let $C(t, s)$ and $c(t, s)$ be defined for $t > s$ as follows

and let $\Gamma:\{(x, t;y, s)\in\mathbb{R}^{2N+2}:(x, t)\not = (y, s)\}\to\mathbb{R}$ be defined by:

Then, the following properties hold:

i) $\Gamma$ is continuous and it is of class $C^\infty$ with respect to the $x$ and $y$ variables, moreover $\partial_x^\alpha\partial_y^\beta\Gamma$ is continuous for any multiindices $\alpha$ and $\beta$;

ii) $\Gamma$ and $\partial_x^\alpha\partial_y^\beta\Gamma$ are Lipschitz continuous with respect to $t$ and with respect to $s$ in $\{(x, t;y, s)\in \mathbb{R}^{2N+2}: a\leq s, t\leq b\quad\mathit{\text{and}}\quad t-s > \delta\}$ for any fixed $a, b\in \mathbb{R}$ and $\delta > 0$;

iii) For any $(y, s)\in \mathbb{R}^{N+1}$ we have:

and for any $g\in C_0(\mathbb{R}^N)$ we also have:

iv) For any $(y, s), (x, t)\in \mathbb{R}^{N+1}$ if $t > s$ we have:

v) For every $(x, t;y, s)\in \mathbb{R}^{2N+2}$ if $t > s$ then:

where $\nu$ is defined in (1.2).

Remark 2.1. It is interesting to notice that this fundamental solution reduces to the previous one when the matrix is constant, indeed in that case we have (see ^[7,8]):

Actually, the assumptions of ^[4] are more general than those we are considering, in particular, with few changes in the statement of Theorem 2.2, we could assume that the structure of the matrix $\mathbb{B}$ is (1.6) instead of (1.3), however, in the more general case (2.1) is not valid. We remark also that ^[4] contains existence and uniqueness for the following Cauchy problem:

As previously mentioned, the theory of ^[2,3] is fundamental in order to prove the main result of this article. Hence we end this section with some known results from the mentioned articles. Coherently with ^[2,3] for any multi-index $\boldsymbol{\alpha}\in \mathbb{N}^N$ we define:

where the coefficients $\{q_i\}_{i = 1}^N$ are the exponents in the dilations $D_0(\lambda)$ (see Definition 1.1).

Theorem 2.3. [3, Theorem 1.4] Under the assumptions stated above the following properties hold:

i) Estimates on $\Gamma$: Let $\boldsymbol{\alpha}\in \mathbb{N}^{N}$ be a multi-index. Then, there exist $c = c(\nu, \boldsymbol{\alpha}) > 0$ and a constant $c_1 > 0$, independent of $\nu$ and $\boldsymbol{\alpha}$, such that:

for every $(x, t), (y, s)\in \mathbb{R}^{N+1}$ with $t\not = s$.

ii) Given $\boldsymbol{\alpha}\in \mathbb{N}^N$ a nonzero multi-index, if $t > s$ we have:

iii) There exist absolute constants $c$, $\mu > 0$ such that, for every fixed $T\in\mathbb{R}$ and every $f\in\mathcal{D}(S_{T})$, $x\in\mathbb{R}^{N}$ and $\tau < t < T$, we have:

iv) {Representation formulas for solutions}: Let $T > 0$ be fixed and let $u\in S^0(S_T)$. If $u\equiv0$ in $\{(x, t):t < 0\}$, then the following representation formula holds:

Moreover, for $i\in \{1, \dots, q\}$, we have the estimate $\Vert \partial_{x_i}u\Vert_{L^\infty(\mathbb{R}^N)}\leq c \Vert \mathcal{L}u\Vert_{L^\infty(S_T)}$ and

v) Estimates: Let $T > \tau > -\infty$. Then, there exist $c, \mu > 0$, only depending on $\nu$ and $\mathbb{B}$, such that, for every $u\in\mathcal{S}^{0}(S_T)$ satisfying $u\equiv0$ in $\{(x, t):t < \tau\}$ and $\mathcal{L}u\in\mathcal{D}(S_{T})$ we have:

and for any $r > 0$ the following inequality holds:

Remark 2.2. We must notice that we are exploiting only a part of the results contained in ^[3]. For instance, similar estimates hold also for a more general class of operators with coefficients $a_{ij}$ which are partially log-Dini continuous and moreover local estimates involving time and space are available for the second order derivatives with respect to the first $q$ space variables (hence they are actually continuous).

From the representation formula above (point ⅳ)) we can easily obtain the following representation formula which entails uniqueness of the solution to the Cauchy problem.

Corollary 2.1. Let $T > 0$ and let $u\in \mathbb{S}(0, T)$ (see Definition 5) be such that $\mathcal{L}u\in L^\infty(\mathbb{R}^N\times(0, T))$. Then, for any $(x, t)\in \mathbb{R}^N\times(0, T]$ we have:

Proof. For $\varepsilon > 0$ let $\psi_\varepsilon:\mathbb{R}\to\mathbb{R}$ be a mollifier with support contained in $(0, \varepsilon)$ and let $\psi^\tau_\varepsilon(t): = \psi_\varepsilon(t-\tau)$, $\phi_\varepsilon^\tau(t): = \int_{-\infty} ^t\psi_\varepsilon^\tau(s)ds$ where $\tau > 0$. The function $\phi_\varepsilon^\tau u$ belongs to $S^0(S_T)$ hence:

Therefore, from $\mathcal{L}(\phi_\varepsilon^\tau u) = -(\partial_t \phi_\varepsilon^\tau)u+ \phi_\varepsilon^\tau \mathcal{L}u = -\psi_\varepsilon^\tau u+\phi_\varepsilon^\tau \mathcal{L}u$, we obtain:

We claim that for $t > \tau$,

Indeed (2.4) follows by dominated convergence while (2.5) is immediate if we observe that $s\mapsto\int_{\mathbb{R}^N}\Gamma(x, t;y, s)u(y, s)dy$ is continuous in $(0, t)$ for any fixed $(x, t)$. We have proved that for any $(x, t)\in \mathbb{R}^N\times (\tau, T)$

The last step is to take the limit for $\tau\to0$. It is apparent that the first term in the right-hand side converges to $\int_0^t\int_{\mathbb{R}^N}\Gamma(x, t;y, s)\mathcal{L}u(y, s)dyds$ while we claim that:

To prove our claim we first notice that for any fixed $(x, t;y)\in \mathbb{R}^N\times(0, +\infty)\times\mathbb{R}^N$ the function

is continuous. Moreover, for any fixed $(x, t)\in \mathbb{R}^N\times(0, +\infty)$ also the function

is continuous and since it tends to zero as $\vert y\vert \to +\infty$ (uniformly in $\tau$) it is also bounded. Therefore for any fixed $(x, t)\in \mathbb{R}^N\times(0, +\infty)$ we have:

Applying the dominated convergence theorem we obtain (2.6). □

3.   Existence for $C_0^\infty$ datum

In this section we shall prove the following theorem:

Theorem 3.1. If $f\in C_0^\infty(S_T)$ then, $u:S_T\to\mathbb{R}$ defined by:

is such that $u\in S^0(S_T)$ and satisfies $\mathcal{L} u = f$.

To prove the above result we need some preliminary lemmas.

Lemma 3.1. Let $\Omega$ be an open set of $\mathbb{R}^N$ and let $w\in C(\Omega)$. If $w$ satisfies the following conditions for some $i\in\{1, \dots, N\}$:

i) For any $K\subset\subset\Omega$

ii) For almost every $x\in \Omega$

Then, for every $K\subset\subset\Omega$

and therefore

where the symbols $D_{x_i} w$ and $\partial_{x_i}w$ denote the weak and the classical derivative of $w$ with respect to $x_i$.

Proof. Let $K$ be an open subset of $\Omega$ with compact closure and let $h_0 > 0$ be such that:

Thanks to the dominated convergence theorem, we obtain:

and since

we have $\partial_{x_i} w = D_{x_i} w$. □

Lemma 3.2. Let $f\in C_0^\infty(S_T)$ and let $\varepsilon > 0$. Moreover, let $u_\varepsilon:S_T\to\mathbb{R}$ be defined by:

Then, we have

Proof. First we observe that $\Gamma$, $\partial_{x_i} \Gamma$ and $\partial_{x_ix_j}^2 \Gamma$ are uniformly bounded on

Indeed, from point ⅰ) of Theorem 2.3 for any $(x, t;y, s)\in S_T\times S_T$ if $t-s > \varepsilon$ we have:

for some fixed constants $a, a' > 0$. Now we claim that a similar bound on $\partial_t \Gamma(x, t;y, s)$ holds in every set of the kind:

with $K\subset\subset\mathbb{R}^N$. Actually, form point ⅲ) of Theorem 2.2, for almost any

such that $t-s > \varepsilon$ we have $\mathcal{L}\Gamma(x, t;y, s) = 0$, taking $a$ and $a'$ as before:

With these preliminary observations, we can proceed with the computation of the classical derivative of $\Gamma$ with respect the variable $t$. For $|h|\in(0, \varepsilon/2)$ we compute the incremental ratio:

We want to prove that for a.e. $(x, t)\in S_T$

First we consider $B_h$. Thanks to the estimates on $\partial_t\Gamma$, we have

which tends to zero as $h\to0$. The convergence of $C_h$ is easily obtained. Indeed, owing to the mean value theorem, it follows that for any $h$ there exists $\delta = \delta(h)\in(0, 1)$ such that

Hence, taking the limit as $h\to 0$, by dominated convergence, we obtain

It is left to prove

Since the derivative $\partial_t\Gamma$ exists a.e. then, for a.e. $(x, t)\in S_T$ and a.e. $(y, s)\in S_T$, we have

and moreover, thanks to the estimate on $\partial_t\Gamma$, for a.e. $(x, t)\in S_T$, we also have

Therefore, applying the dominated convergence, we finally obtain that for a.e. $(x, t)\in S_T$:

This proves that for a.e. \((x, t)\in S_T\)

Now we shall obtain that the classical derivative (which is defined almost everywhere) is also a weak derivative by observing that the incremental ratio is uniformly locally bounded. Actually, let $K$ be a fixed compact subset of $\mathbb{R}^N$, then from the estimates obtained at the beginning of the proof, for any $(x, t)\in K\times (-\infty, T)$ we derive the following estimates:

and

The last inequality follows from point ⅳ) of Theorem 2.2. By Lemma 3.1 we can conclude that for almost any $(x, t)$ the partial derivative with respect to $t$ exists:

and moreover it is also a weak derivative.

For the derivatives with respect to the variables $x_i$ for $i\in\{1, \dots, N\}$ we can apply the standard theorem of differentiation under the integral. Indeed for any fixed $t\in (-\infty, T)$ the function

is of class $C^2$ with respect to $x$ and moreover it and its $x$-derivatives are uniformly bounded by an $L^1$ function since for any $(x, y, s)\in\mathbb{R}^N\times\mathbb{R}^N\times(-\infty, t-\varepsilon)$, we have

Therefore, applying the standard theorem of differentiation under the integral sign we get that the classical derivatives of $u_\varepsilon$ exist for all $(x, t)\in S_T$ and moreover

Since the integrands are continuous and uniformly bounded by an $L^1$ function by the dominated convergence theorem it follows that $\partial_{x_i} u_\varepsilon$ and $\partial_{x_ix_j}^2u_\varepsilon$ are continuous, hence these derivatives are also weak derivatives. Finally, exploiting (3.1)–(3.3), we get that for almost every $(x, t)\in S_T$

Hence, applying point ⅱ) of Theorem 2.2, it follows:

□

Lemma 3.3. If $f\in C_0^\infty(S_T)$ then, for every $K\subset\subset\mathbb{R}^N$

Proof. We proceed as in the first part of the proof of Proposition 3.10 in ^[2]. Owing to points ⅲ) and ⅳ) of Theorem 2.2:

Now we make the change of variable $\{z = D_{0}(\frac{1}{\sqrt{\varepsilon}})(x-E(\varepsilon)y), dz = \frac{1}{\varepsilon^\frac{Q}{2}}dy\}$

which, for $x$ varying in a compact set, vanishes uniformly as $\varepsilon\to0$. □ We can finally prove Theorem 3.1.

Proof of Theorem 3.1. In order to prove the existence theorem we exploit the uniform convergence of $u_\varepsilon$ and its derivatives.

We begin with the convergence of $u_\varepsilon$.

By point ⅳ) of Theorem 2.2 we easily get:

Then for the first derivatives we proceed in the same way as in Corollary 3.12 in ^[2]. For any $i\in \{1, \dots, q\}$ we have:

Now we claim that the integral

is absolutely convergent and that $\partial_{x_{i}x_{j}}u_{\varepsilon}$ converges uniformly to it. Indeed, since $f$ is $C_0^\infty(\mathbb{R}^{N+1})$ (therefore also $\mathcal{D}(\mathbb{R}^{N+1})$), for any$(x, t)\in S_T$ there exists $\tau < t$ such that ${\rm{supp}}(f)\subset(\tau, +\infty)\times\mathbb{R}^N$, hence, owing to point ⅲ) of Theorem 2.3, we have:

Moreover, thanks to Theorem 2.3 point ⅱ) we have:

Therefore, from point ⅲ) of Theorem 2.3, we obtain:

We have proved that $u$ has continuous derivatives up to the second order with respect to the variables $x_i$ for $i\in\{1, \dots, q\}$. Applying Lemma 3 we obtain that $\mathcal{L}u_\varepsilon\to f$ in $L^\infty_{loc}$ hence thanks to the formerly proved limits we get:

The convergence is also in $D'(S_T)$, so

in $D'(S_T).$ Therefore $Yu\in L^\infty(S_T)$ and $\mathcal{L}u = f$. Thus $u\in S^0(S_T)$. □

4.   Existence of a solution under minimal assumptions on the datum

This section is devoted to the proof of Theorem 1.1. As remarked in Section 1 the main point in the proof of this theorem is the existence of a solution, since uniqueness and the stability estimates follow from the results of ^[2,3].

Theorem 4.1. If $f\in \mathcal{D}(S_T)$ and is compactly supported, then, the function $u:S_T\to\mathbb{R}$ defined by:

is such that $u\in S^0(S_T)$, $\mathcal{L}u = f$ and moreover we have the stability estimates (2.2) and (2.3).

Proof. In this proof the symbol $*$ denotes the standard convolution.

Let $f_\varepsilon = f*\varphi_\varepsilon$ be the convolution of $f$ with a compactly supported mollifier $\varphi_\varepsilon:\mathbb{R}^{N+1}\to \mathbb{R}$ and let $u_\varepsilon\in S^0(S_T)$ be the solution given by the existence Theorem 3.1 with datum $f_\varepsilon$, namely:

We want to prove that $u_\varepsilon \mathop {\rightharpoonup}\limits_\varepsilon ^* u$ in $L^\infty(S_T)$. First we prove that

Indeed, since for any $\phi\in L^1(\mathbb{R}^{N+1})$, $\phi*\bar\varphi_\varepsilon\xrightarrow[\varepsilon]{L^1(S_T)}\phi$ where $\bar\varphi_{\varepsilon}(x, t): = \varphi_{\varepsilon}(-x, -t)$, then we easily have

Now, since $f$ has compact support, there exist $T_1 < T$ such that $supp(f), supp(f_\varepsilon)\subset\mathbb{R}^N\times(T_1, T)$ for any $\varepsilon$ sufficiently small. Then, observing that $\Gamma(x, t;\cdot)\in L^1(\mathbb{R}^N\times (T_1, T))$ for any $(x, t)$, we obtain

which means that $u_\varepsilon(x, t)\to u(x, t)$ for any $(x, t)\in S_T$.

Now, let $\phi\in L^1(S_T)$, since $u_\varepsilon\to u$ pointwise and

applying again the dominated convergence theorem we get:

hence

Then, notice that $\omega_{f_\varepsilon, S_T}\leq\omega_{f, S_T}$ and $\Vert u_\varepsilon\Vert_{L^\infty(S_T)} \leq \Vert f\Vert_{L^\infty(S_T)}$, hence, by points ⅳ), ⅴ) of [Theorem 2.3], it follows that, for some fixed constant $c > 0$ for any $i, j\in\{1, \dots, q\}$ and any $\varepsilon > 0$ sufficiently small we have

and

Therefore the $L^\infty(S_T)$ norms of $u_\varepsilon$, $\partial_{x_i}u_\varepsilon$ ($i\in\{1, \dots, q\}$), $\partial_{x_ix_j}^2u_\varepsilon$ ($i, j\in\{1, \dots q\}$) and $Yu_\varepsilon$ are uniformly bounded (w.r.t. $\varepsilon$). Hence we apply the Banach-Alaoglu theorem (in $L^\infty(S_T)$) to $u_\varepsilon$, $\partial_{x_i}u_\varepsilon$, $\partial_{x_ix_j}^2u_\varepsilon$ and $Yu_\varepsilon$. In this way we obtain a sequence of real numbers $\varepsilon_k\downarrow0$ such that $u_{\varepsilon_k}$ and $\partial_{x_i} u_{\varepsilon_k}$, $\partial_{x_ix_j}^2u_{\varepsilon_k}$ and $Yu_{\varepsilon_k}$ converge in the weak topology $\sigma(L^\infty(S_T), L^1(S_T))$ to functions in $L^\infty(S_T)$. Notice that $u_{\varepsilon_k}$, $\partial_{x_i} u_{\varepsilon_k}$, $Yu_{\varepsilon_k}$ and $\partial_{x_ix_j}^2u_{\varepsilon_k}$ converge also in the sense of distributions hence, thanks to the uniqueness of the limit in the sense of distributions, we obtain that $u\in S^0(S_T)$ and for $i, j\in\{1, \dots, q\}$

in $L^\infty(S_T)$. Notice that actually it is not necessary to take a subsequence since the limit is unique.

It is left to show that $\mathcal{L}u = f$ but thanks to (4.1) we only need to prove that $\mathcal{L}u_\varepsilon \mathop {\rightharpoonup}\limits_\varepsilon^* \mathcal{L}u$. Let $\phi\in L^1(S_T)$ and $i, j\in\{1, \dots, q\}$ then:

Notice that the estimates are valid since point ⅴ) of Theorem 2.3 only require $u\in S^0(S_T)$ therefore the proof is completed. □

Theorem 4.2. Let $f\in D(S_T)$ such that $supp(f)\subset \mathbb{R}^N\times(\tau, T)$ for some $\tau\in(-\infty, T)$ and let $u:S_T\to\mathbb{R}$ be defined by:

Then, $u\in S^0(S_T)$, $\mathcal{L}u = f$ and we have the estimates (2.2) and (2.3).

Proof. The proof of this theorem is similar to the previous one but this time $f$ is approximated in a different way. Let $\{\phi_i\}_i\subset C_0^\infty(\mathbb{R}^N)$ be such that $0\leq\phi_i\uparrow1$ (as $i\to+\infty$), define $f_i = f \phi_i$ and $u_i$ to be the solution of $\mathcal{L}u_i = f_i$ given in Theorem 4.1. Notice that $u_i$ admits the representation formula:

Since $\Gamma(x, t;\cdot)$ is integrable for any fixed $(x, t)$, thanks to dominated convergence, $u_i\to u$ pointwise. Moreover $\Vert u_i-u\Vert_{L^\infty(S_T)} \leq T\Vert f-f_j\Vert_{L^\infty(S_T)} \leq T\Vert f\Vert_{L^\infty(S_T)}$ hence, taking any $\psi\in L^1(S_T)$, by dominated convergence we obtain:

thus $u_i\mathop {\rightharpoonup}\limits_{}^*u$ in $L^\infty(S_T)$.

Now we observe that $f_i \mathop {\rightharpoonup}\limits_i^* f$ in $L^\infty(S_T)$. Indeed $\Vert f_i-f\Vert_{L^\infty(S_T)} \leq\Vert f\Vert_{L^\infty(S_T)}$ hence, for fixed $\psi\in L^1(S_T)$, we can apply again the dominated convergence obtaining:

Finally, thanks to the estimates Theorem 2.3 $v$) employing the same argument of the proof of Theorem 4.1 we find that $u\in S^0(S_T)$ satisfies the estimates (2.2), (2.3) and for $k, j\in\{1, \dots, q\}$

in $L^\infty(S_T)$. This entails also $\mathcal{L}u = f$. □ In order to complete the proof of Theorem 1.1 we would like to sum the solution of the homogeneous problem from ^[4] and the solution of the nonhomogeneous problem given Theorem 4.2, hence some remarks on the solution of the homogeneous problem are in order.

Remark 4.1. As already mentioned, the paper ^[4] contains the solution to the problem:

Moreover, the explicit form of the solution is the following:

It is easily seen that $u$ has continuous derivatives of any order with respect to $x$ while it is locally Lipschitz with respect to $t$ in $(0, +\infty)\times \mathbb{R}^N$. Moreover $u$ satisfies $\mathcal{L}u(x, t) = 0$ for all $x\in\mathbb{R}^N$ and a.e. $t\in (0, +\infty)$ therefore $u\in \mathbb{S}(0, +\infty)$ and $\mathcal{L}u = 0$ in the sense of distributions.

Now we can conclude the proof of Theorem 1.1.

Proof of Theorem 1.1. Let $u_f$ be the solution of the problem (1.7) with datum $g\equiv0$ while let $u_g$ be the solution to (1.7) with $f\equiv0$. From what we have seen until now the function $u: = u_f+u_g$ satisfies $u\in \mathbb{S}(0, +\infty)$, $\mathcal{L}u = f$ and $u(\cdot, 0) = g(\cdot)$ and moreover it is the unique solution of (1.7) and is given by

We are left to prove the estimates (1.9) and (1.10), to this aim we proceed on $u_f$ and $u_g$ separately Concerning $u_f$ applyig (2.2) and (2.3) we easily obtain:

While to prove the estimates for $u_g$ we shall represent $u_g$ as a solution of a nonhomogeneous problem (with null initial datum at $t = -1$) so that we can exploit again the estimates (2.2) and (2.3). Let $\varphi\in C^\infty(\mathbb{R})$ be a cut-off function satisfying $\varphi(t) = 0$ for $t\leq-1$ and $\varphi(t) = 1$ for $t\geq0$. From the regularity properties of $g$ it is easily verified that $g\otimes\varphi(\cdot, \cdot+1)$ belongs to $\mathbb{S}(0, +\infty)$ and therefore thanks to Corollary 2.1 we can represent $g\otimes\varphi$ as:

In particular for $t = 0$ we have:

Moreover, it is easily seen that the function $v$ defined by:

is a solution of:

Thanks to the uniqueness of the solution, from (4.7) and (4.6) we obtain

Hence we can apply the estimates (2.2) and (2.3) to $v$ (with $\tau = -1$) in order to obtain some estimates for $u_g$. In particular we have:

In order to conclude the proof we only need to bound the quatities on the right hand side of (4.8) and (4.9) in terms of $g$. Computing explicitly $\mathcal{L}(g\otimes \varphi)$ we obtain:

From these computations we easily infer the following estimates:

where the constant $k$ depend only on $\Vert a_{ij}\Vert_{L^\infty(\mathbb{R}^{N+1})}$, $\Vert \varphi\Vert_{L^\infty(\mathbb{R})}$ and $\Vert \varphi'\Vert_{L^\infty(\mathbb{R})}$. Thanks to (4.8)–(4.11) we obtain:

Finally, combining (4.4), (4.5), (4.12) and (4.13) we deduce (1.9) and (1.10). □

5.   Conclusions

Starting from the theory of ^[2,3,4] we were able to prove the well-posedness of a Cauchy problem for a degenerate Kolmogorov-Fokker-Planck operator under weak regularity assumptions on the coefficients and an the data. The main point was to prove the existence of a solution for the problem with null initial data since the existence of a solution to the homogeneous problem with nonnull initial data, the stability estimates and the uniqueness of the solution follows from known results. Since an explicit fundamental solution is known, in order to find a solution to the Cauchy problem we employed the Duhamel method. However, due to the low regularity of the coefficients, we first considered the case of smooth and compactly supported datum, then, by approximation, we have shown the existence of a solution for a compactly supported datum satisfying the minimal regularity assumptions and finally we weakened the assumptions on the support of the datum.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper originates from the master thesis ^[1] done at Politecnico di Milano under the guidance of Prof. Marco Bramanti and Dr. Stefano Biagi to whom my gratitude goes. The mentioned thesis is focused on the same operator but is based on the Schauder theory hence after suitable modifications the Sections 3 and 4 are taken from Sections 2.2 and 2.3 of ^[1]. I wish to thank Prof. Marco Bramanti and Dr. Stefano Biagi, for the help and the many suggestions they gave me during the writing of this article.

Conflict of interest

The author declares no conflict of interest.