Existence of a solution for this hypoelliptic-alike PDE

Question

I know that this question may result rater vague and somehow out of context, still I am hoping you could help me. Assume we have the following equation

\begin{align} \boxed{\partial_t u(t,x,z)=\mathcal L(t,x) u(t,x,z)+h(t,x)\partial_z u(t,x,z)} \end{align}

where \begin{align} \mathcal L(t,x)=\sum_{i=1}^{d}\sum_{j=1}^{d} a_{ij}(t,x)\frac{\partial^2}{\partial x_j \partial x_i}+\sum_{i=1}^{d} b_i(t,x)\frac{\partial}{\partial x_i}+c(t,x). \end{align}

where the $a$'s an the $b$'s are very well behaved functions (we can assume they are $C^{\infty}$, bounded, and with bounded derivatives) and we can assume that $a_{ij}$ is uniformly elliptic.

The fact that there's no second order derivative with respect to $z$ makes me think that we are dealing with the prototipe of "hypoelliptic" PDEs.

The problem is that I am not really familiar with Hörmander's results, and I find particularly difficult to read and understand his papers. I was wondering under which assumptions regarding $h$ can we conclude that the latter equations has a solution? Is is possible to show existence even though we don't assume the necessary conditions for hypoellipticity?

Thanks in advance and please let me know if something is not clear.

Answer 1

Too long for a comment. This is a linear equation $$ \partial_t-\mathcal L(t,x,\partial_x)-h(t,x) \partial_z $$ and since the coefficients of $\mathcal L$ do not depend on $z$, you can Fourier transform with respect to $z$ and get the equation with the real parameter $\zeta$ given by the operator $$ \partial_t-\mathcal L(t,x,\partial_x)-h(t,x) i\zeta. $$ Assuming that the operator $\mathcal L$ is elliptic, you get a parabolic equation with parameter $\zeta$ and if you assume that $h$ is real-valued, you may check Avner Friedman's book on parabolic equations to get some theoretical information.

Answer 2

Even this answer is perhaps a very long comment. A larger class of equations, strictly including the class of hypoelliptic ones, was first studied by Mstislav Vsevolodovich Keldysh, Gaetano Fichera and later by Ol’ga Arsen’evna Oleĭnik and Evgeniĭ Vladimirovich Radkevič: this is precisely the class of equations with nonnegative characteristics. This class includes (standard and degenerate) elliptic and parabolic PDE, and others kind of PDEs as well: it was at first investigated with the aim to develop a unifying approach to the general theory of PDES, precisely to develop a general theory of variable coefficients linear differential operators. As such, the PDEs studied are not required to strictly have $C^\infty$-regular coefficients, and no other requirements are imposed on coefficients except for the positive semi-definiteness condition on the leading coefficient matrix, i.e. $$ \sum_{i,j=1}^n a_{ij}\xi_i\xi_k \ge 0 \quad\forall \boldsymbol{\xi}=(\xi_1, \ldots,\xi_n)\in\Bbb R^n \label{1}\tag{1} $$ And the above equation is obviously an equation with non positive characteristics, since by putting $$ (\tilde a_{ij})_{i,j=1,\ldots,d+1}=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1d} & 0\\ a_{21} & a_{22} & \cdots & a_2^n & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{d1} & a_{d2} & \cdots & a_{dd} & 0 \\ 0 & 0 &\cdots & 0 & 0 \end{pmatrix} $$ we have that \eqref{1} is trivially satisfied since $(a_{ij})_{i,j=1,\ldots,d}$ is uniformly elliptic and $$ \sum_{i,j=1}^{d+1} \tilde a_{ij}\xi_i\xi_k=0 $$ for all vectors of the form $\boldsymbol{\xi}=(\xi_1, \ldots,\xi_d, \zeta)=(0, \ldots, 0, \zeta)\in\Bbb R^{d+1}$, with obvious meaning for $\zeta$.

Now, even if we try to sketch the basic theory of such classe of equations, this would led us too far away: the theory involves non only (and at a later stage) Hörmanders's condition but also Fichera's function and other interesting and important techniques. Also, several existence and uniqueness theorems for weak as well classical solutions are stated and proved by the authors above and various other ones, therefore I advice you to have look at the books and papers cited in the "References section".

Notes

• Reference [1] is the main monograph on the topic: it offers a presentation of the state of the theory up to 1973, including all mayor results in the area, therefore it is possible that you'll find something useful there.
• The last two papers [2] and [3] by Radkevič form an updated and comprehensive survey describing the state of topic up to 2008, so perhaps they would be the best places where to start if you want to have a "taste" of the current theory.

Again my two cents.

References

[1] Ol’ga Arsen’evna Oleĭnik, Evgeniĭ Vladimirovich Radkevič, "Second order equations with nonnegative characteristic form", (Russian) Itogi Nauki i Tekhniki. Seriya "Matematicheskii Analiz" 1969, 7-252 (1971), Zbl 0217.41502. Translated as the book:
Ol’ga Arsen’evna Oleĭnik, Evgeniĭ Vladimirovich Radkevič, Second order equations with nonnegative characteristic form, Translated from the Russian by Paul C. Fife. New York-London: Plenum Press, 1973, pp VII+259, ISBN: 0-306-30751-0, MR0457908.

[2] Evgeniĭ Vladimirovich Radkevich, "Equations with nonnegative characteristic form. I", (English. Russian original) Journal of Mathematical Sciences, New York 158, No. 3, 297-452 (2009); Translation from Sovrem. Mat. Prilozh. 55 (2008), DOI:10.1007/s10958-009-9395-1, MR2675371, Zbl 1200.35158.

[3] Evgeniĭ Vladimirovich Radkevich, "Equations with nonnegative characteristic form. II", (English. Russian original) Journal of Mathematical Sciences, New York 158, No. 4, 453-604 (2009); translation from Sovrem. Mat. Prilozh. 56 (2008), DOI:10.1007/s10958-009-9395-1, MR2675371, Zbl 1200.35157,