Nonlinear partial differential equations on noncommutative Euclidean spaces

Abstract

Noncommutative Euclidean spaces—otherwise known as Moyal spaces or quantum Euclidean spaces—are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the analysis of some equations radically simplifies in the strictly noncommutative setting.

Appendix A. Existence theorems for parabolic differential equations

In this paper, we have used the theory of abstract Cauchy problems to deal with existence and uniqueness of partial differential equations on ${\mathbb {R}}^d_\theta$. The general theory of abstract Cauchy problems is covered in, e.g. [41, Chapter 4], [2, Section 1.2], [3, Chapter 3], [51, Chapter 5], [34, Appendix G], [64, 22, Chapter 7].

We have made use of a “blow-up” criterion (Theorem 7.3) which is certainly not novel but for which we have been unable to find a reference.

Proof of Theorem 7.3

Suppose that $T_{x_0} < \infty$. If this is the case, then we must have $\Vert x(s)\Vert _{Y} \rightarrow \infty$ as $s\rightarrow T_{x_0}$. Indeed, otherwise there would exist $R > 0$ and a sequence $\{\varepsilon _j\}_{j=0}^\infty$ with $\varepsilon _j\rightarrow 0$ such that $\Vert x(T_{x_0}-\varepsilon _j)\Vert _Y \le R$. According to Theorem 7.1, there exists $T_R$ such that the solution can then be extended to an interval $[0,T_{x_0}-\varepsilon _j+T_R)$. This contradicts the maximality of $T_{x_0}$ when j is large enough, and hence, if $T_{x_0}<\infty$ then

$$\begin{aligned} \sup _{0<t<T_{x_0}}\Vert x(t)\Vert _{Y} = \infty . \end{aligned}$$

The assumption of the theorem is that there exists $0< C_{x_0} < \infty$ such that

$$\begin{aligned} \Vert F(x(s))\Vert _{X} \le C_{x_0}\Vert x(s)\Vert _Y,\quad 0\le s < T_{x_0}. \end{aligned}$$

Let $0< r< t < T_{x_0}$. Then by definition, we have

$$\begin{aligned} x(t) = e^{tL}x_0 + \int _0^t e^{(t-s)L}F(x(s))\,\textrm{d}s. \end{aligned}$$

Using the contractivity of the semigroup $e^{tL}$ and Lemma 7.2, it follows that

$$\begin{aligned} \Vert x(t)\Vert _Y \le \Vert x_0\Vert _Y+\int _0^{t} \Vert e^{(t-s)L}\Vert _{X\rightarrow Y}\Vert x(s)\Vert _Y \,\textrm{d}s. \end{aligned}$$

By Gronwall’s inequality

$$\begin{aligned} \Vert x(t)\Vert _Y \le \exp (\int _0^{t}\Vert e^{sL}\Vert _{X\rightarrow Y}\,\textrm{d}s)\Vert x_0\Vert _{Y} \le \exp (Ct^{1-\gamma })\Vert x_0\Vert _Y. \end{aligned}$$

Therefore, if $T_{x_0}<\infty ,$ we have

$$\begin{aligned} \sup _{0< t< T_{x_0}} \Vert x(t)\Vert _{Y} \le \exp (CT_{x_0}^{1-\gamma })\Vert x_0\Vert _Y <\infty . \end{aligned}$$

which is a contradiction. $\square$