Density of the futures of null solutions for spatially tempered linear constant-coefficient PDEs (Sasane 2021, Section 7)

Section 7 of Sasane, 'On the existence of spatially tempered null solutions to linear constant coefficient PDEs' (Israel Journal of Mathematics 244 (2021), 273-291), asks, verbatim, 'For which p is the set of futures of null solutions dense in the set of futures of all solutions?' In the spatially tempered solution class introduced there the null-solution space itself has been characterized, but this density question was explicitly left open. It is the direct generalization, to that spatially tempered class, of the density theorem proved by Hörmander for smooth null solutions in 'Null solutions of partial differential equations' (1959). The task is to answer it.

Background

A linear partial differential operator with constant coefficients on space-time $\mathbb{R}^{d+1}=\mathbb{R}^d_x\times\mathbb{R}_t$ is obtained from a polynomial $p\in\mathbb{C}[X_1,\ldots,X_d,T]$ by substituting $\partial/\partial x_k$ for each $X_k$ and $\partial/\partial t$ for $T$; the resulting operator is denoted $D_p$. A null solution is a distributional solution of $D_p u=0$ whose past is zero, $u|_{t<0}=0$; the existence of a nontrivial null solution is exactly the failure of uniqueness for the Cauchy problem with zero initial data on the hyperplane $t=0$. The existence theory goes back to Hörmander's 1955 Acta paper and to Hörmander's 1959 'Null solutions of partial differential equations': a nontrivial null solution exists precisely when the initial hyperplane is characteristic for $D_p$, that is, when the principal part $p_m$ (the highest-total-degree homogeneous part of $p$) vanishes at the time normal. In the formulation of Sasane (2021, Proposition 1.4, following Theorems 8.6.7-8.6.8 of Hörmander's treatise ALPDO I), for $S=C^\infty(\mathbb{R}^{d+1})$ or $S=\mathcal{D}'(\mathbb{R}^{d+1})$ (smooth functions, or distributions) the null-solution space $N_S(p)$ is trivial if and only if $\deg p=\deg p(0,T)$, where $p(0,T)$ is obtained from $p$ by setting all spatial variables to zero and $\deg$ denotes total degree. The 1959 paper goes beyond existence and proves closure theorems for the span of null solutions; in particular, from it one obtains that the futures (restrictions to $\{t>0\}$) of smooth null solutions are dense in the futures of all smooth solutions whenever every irreducible factor $p_1$ of $p$ satisfies $\deg p_1\neq\deg p_1(0,T)$ (as recorded in Section 7 of Sasane 2021).

The classical category is physically too permissive: for the diffusion equation, $p=T-(X_1^2+\cdots+X_d^2)$, one has $\deg p=2\neq 1=\deg p(0,T)$, and pathological null solutions with spatial profiles growing faster than $e^{|x|^2}$ exist in $C^\infty$ and $\mathcal{D}'$, even though a substance density that is zero up to time $0$ should remain zero. Sasane (2021) therefore introduced solution classes with a spatial growth restriction and no growth restriction in time. The central one is the spatially tempered class $S=L(\mathcal{D}(\mathbb{R}),\mathcal{S}'(\mathbb{R}^d))$: the continuous linear maps from $\mathcal{D}(\mathbb{R})$ (compactly supported smooth functions of time) to $\mathcal{S}'(\mathbb{R}^d)$ (tempered distributions, that is, distributions of at most polynomial growth in the spatial variables), with the topology of pointwise convergence; via the Schwartz kernel theorem these are exactly the distributions on $\mathbb{R}^{d+1}$ whose spatial profile at each time is tempered. The question also has a systems-theoretic meaning: in behavioral systems theory (the Willems school), a system is identified with its space of trajectories, and the dichotomy between time-autonomy (every trajectory determined by its past, equivalently a trivial null-solution space) and time-controllability (trajectories can be concatenated in time) was introduced for two-dimensional constant-coefficient behaviors by Sasane, Thomas, and Willems (2002), with algebraic characterizations for spatially tempered two-dimensional behaviors given by Sasane (2004).

For the spatially tempered class, Sasane (2021) completed the existence side: writing $p=a_0+a_1T+\cdots+a_nT^n$ with $a_j\in\mathbb{C}[X]$, the $X$-content $C_X(p)$ is the ideal in $\mathbb{C}[X]$ generated by the coefficients $a_j$, and $V(\cdot)$ denotes the common complex zero set, the null-solution space satisfies $N_S(p)=\{0\}$ if and only if $V(C_X(p))\cap i\mathbb{R}^d=\emptyset$, where $i\mathbb{R}^d$ is the set of purely imaginary vectors (Theorems 1.9 and 4.1, stated as Theorem 1.10). For the spatially periodic classes $\mathcal{D}'_A(\mathbb{R}^{d+1})$ the analogous criterion reads: $N(p)=\{0\}$ if and only if for every $v\in A^{-1}2\pi\mathbb{Z}^d$ there is $t\in\mathbb{C}$ with $p(iv,t)\neq 0$ (Theorem 6.1). For the remaining classes considered there - spatial profiles restricted to Besov, Sobolev, Schwartz, or compactly supported distributions - the null-solution space is trivial for every $p\neq 0$. The approximation side, however, was left open: Section 7 of Sasane (2021) closes with the question 'For which p is the set of futures of null solutions dense in the set of futures of all solutions?' and leaves the analogous question in the alternative solution spaces for future investigation. That question, in the spatially tempered class (with the spatially periodic class as its sibling), is the primary case addressed here.

Problem Statement

Let $d\geq 1$ and let $p\in\mathbb{C}[X_1,\ldots,X_d,T]$ be a nonzero polynomial; substituting $\partial/\partial x_k$ for $X_k$ (for $k=1,\ldots,d$) and $\partial/\partial t$ for $T$ gives the constant-coefficient operator $D_p$ on $\mathbb{R}^{d+1}=\mathbb{R}^d_x\times\mathbb{R}_t$. Let $S=L(\mathcal{D}(\mathbb{R}),\mathcal{S}'(\mathbb{R}^d))$ be the space of continuous linear maps from $\mathcal{D}(\mathbb{R})$ (compactly supported smooth functions of $t$) to $\mathcal{S}'(\mathbb{R}^d)$ (tempered distributions in $x$), with the topology of pointwise convergence; each $u\in S$ is the distribution on $\mathbb{R}^{d+1}$ determined by $\varphi\mapsto u(\varphi)$, with the spatial derivatives acting by duality on the $\mathcal{S}'(\mathbb{R}^d)$-factor and $\partial/\partial t$ by duality on the time variable. A null solution is a $u\in S$ with $D_p u=0$ and zero past, that is, $u(\varphi)=0\in\mathcal{S}'(\mathbb{R}^d)$ for every $\varphi\in\mathcal{D}(\mathbb{R})$ supported in $(-\infty,0)$; $N_S(p)$ denotes this subspace. The future of $u\in S$ is its restriction $u|_{t>0}\in L(\mathcal{D}((0,\infty)),\mathcal{S}'(\mathbb{R}^d))$ (the same map acting on test functions supported in $(0,\infty)$), again with the topology of pointwise convergence.

Determine for which polynomials $p$ the set of futures of null solutions $\{u|_{t>0}:u\in N_S(p)\}$ is dense, in this topology, in the set of futures of all solutions $\{u|_{t>0}:u\in S,\ D_p u=0\}$; that is, characterize these $p$ - in the spirit of the known triviality criterion $N_S(p)=\{0\}\iff V(C_X(p))\cap i\mathbb{R}^d=\emptyset$, where $C_X(p)\subset\mathbb{C}[X]$ is the ideal generated by the coefficients of $p$ as a polynomial in $T$ and $V(\cdot)$ is its common complex zero set.

Determine the analogous characterization in the spatially periodic class $\mathcal{D}'_A(\mathbb{R}^{d+1})$ of distributions on $\mathbb{R}^{d+1}$ whose spatial part is periodic with period set $\{a_1,\ldots,a_d\}$ (a linearly independent set in $\mathbb{R}^d$; $A$ is the matrix whose rows are $a_1^{\mathsf T},\ldots,a_d^{\mathsf T}$), whose null-solution triviality criterion reads: $N(p)=\{0\}$ if and only if for every $v\in A^{-1}2\pi\mathbb{Z}^d$ there is $t\in\mathbb{C}$ with $p(iv,t)\neq 0$. In the remaining classes of Sasane (2021) - spatial profiles restricted to Besov, Sobolev, Schwartz, or compactly supported distribution spaces - the null-solution space is trivial for every $p\neq 0$, so there the density question reduces to whether every solution has zero future.

References

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