On the Kernel of a Polynomial of Scalar Derivations

Abstract

In this paper, by using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. Moreover, a gradient representation for the associated Cauchy Problem solution is derived.

1. Introduction and Problem Formulation

The gradient-type representations for some solutions, Lie algebras, gradient systems in a Lie algebra, algebraic representation of gradient systems and their integral manifolds, have been studied for a long time, with remarkable results, by Vârsan [1] and Barbu et al. [2]. Apart from the linear higher order PDEs, the characteristic system method is intensively used for solving linear or nonlinear SPDEs and in this respect, we mention Iftimie et al. [3]. For other different but related viewpoints on this subject, the reader is directed to Friedman [4], Sussmann [5], Crandall and Souganidis [6], Sontag [7], Bressan and Shen [8], Evans [9], Brezis [10], Parveen and Akram [11], Treanţă and Vârsan [12], Treanţă [13].

In this paper, by using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. Moreover, a gradient representation for the associated Cauchy Problem solution is derived. As the main motivation of this study, the mathematical framework developed in this work can be extended for the study of some higher-order hyperbolic, parabolic or Hamilton–Jacobi equations involving a finite set of derivations. For instance, a simple m-th order Hamilton–Jacobi equation has the following expression

$${\displaystyle {\left(\overrightarrow{Z}\right)}^m\left(\phi \right)(t,x)=\sum _{k=0}^{m-1}{a}_k\left(t\right){\left(\overrightarrow{Z}\right)}^k\left(\phi \right)(t,x)+f\left(t\right),(t,x)\in [0,T]\times R^n,}$$

(1)

where ${\displaystyle \overrightarrow{Z}:{\mathit{C}}^1\left([0,T]\times R^n;R\right)\to \mathit{C}\left([0,T]\times R^n;R\right)}$ is a linear application defined by

$${\displaystyle \overrightarrow{Z}\left(\phi \right)(t,x)={\partial }_t\phi (t,x)+⟨{\partial }_x\phi (t,x),X(t,x)⟩,t\in [0,T],x\in R^n}$$

(see ${\displaystyle \overrightarrow{Z}}$ as being generated by the vector field ${\displaystyle Z\left(z\right)=col(1,X(z\left)\right),z=(t,x)}$, where ${\displaystyle X\in \left({\mathit{C}}_b^1\cap {\mathit{C}}^m\right)\left([0,T]\times R^n;R^n\right),m\ge 2}$, and the index ${\displaystyle b}$ of ${\displaystyle {\mathit{C}}_b^1}$ is for bounded) and ${\displaystyle \{a_k,f:0\le k\le m-1\}\subseteq \mathit{C}([0,T];R)}$. Using standard notation, ${\displaystyle \phi =y_0,{\left(\overrightarrow{Z}\right)}^k\left(\phi \right)=y_k,0\le k\le m-1}$, rewrite $\left(1\right)$ as a system of Hamilton–Jacobi equations

$${\displaystyle \overrightarrow{Z}\left(y_0\right)(t,x)=y_1(t,x),\cdots ,\overrightarrow{Z}\left(y_{m-2}\right)(t,x)=y_{m-1}(t,x),}$$

(2)

$${\displaystyle \overrightarrow{Z}\left(y_{m-1}\right)(t,x)=\sum _{k=0}^{m-1}{a}_k\left(t\right)y_k(t,x)+f\left(t\right),(t,x)\in [0,T]\times R^n.}$$

A classical solution ${\displaystyle y(t,x)=\left(y_0(t,x),\dots ,y_{m-1}(t,x)\right)\in R^m,(t,x)\in [0,T]\times R^n}$, associated with $\left(2\right)$, means a first order continuously differentiable mapping ${\displaystyle y\in {\mathit{C}}^1\left([0,T]\times R^n;R^m\right)}$ satisfying $\left(2\right)$ for any ${\displaystyle (t,x)\in [0,T]\times R^n}$. The first component of a solution verifying $\left(2\right)$ stands for a classical solution of the higher order Hamilton–Jacobi equation $\left(1\right)$. It is well known that the characteristic system method is associated with the classical solutions of the PDEs (at least continuous functions). In the previous context, a solution of the corresponding higher order PDEs involves a characteristic system containing a bounded variation component as solution for some ODEs.

Throughout this paper, let ${\displaystyle 0\in I\subseteq R}$ be an open interval. Consider a polynomial of the scalar derivation ${\displaystyle \frac{d}{dt}}$,

$${\displaystyle P_m\left(t;\frac{d}{dt}\right)=a_1\left(t\right)+a_2\left(t\right)\left(\frac{d}{dt}\right)+\cdots +a_m\left(t\right){\left(\frac{d}{dt}\right)}^{m-1}-{\left(\frac{d}{dt}\right)}^m,}$$

(3)

where ${\displaystyle m\ge 1,a_j\in L_{\infty }\left(I\right),j\in \left\{1,2,\dots ,m\right\}}$. Define

$${\displaystyle H_{\infty }^m\left(I\right)=\left\{h\in C^{m-1}\left(I\right):{\left(\frac{d}{dt}\right)}^m\left(h\right)\in L_{\infty }\left(I\right)\right\}}$$

(4)

and consider ${\displaystyle Ker\left(P_m\right)\subseteq H_{\infty }^m\left(I\right)}$, where

$${\displaystyle Ker\left(P_m\right)=\left\{h\in H_{\infty }^m\left(I\right):P_m\left(t;\frac{d}{dt}\right)\left(h\right)\left(t\right)=0,a.e.t\in I\right\}.}$$

(5)

The procedure of characteristic systems (see Friedman [4], Vârsan [1]) allows us to describe ${\displaystyle Ker\left(P_m\right)}$ by solving Cauchy Problems for the corresponding system of ODEs using a vector variable

$${\displaystyle y=col\left(y_1,y_2,\dots ,y_m\right),\frac{dy}{dt}=Ay+\sum _{i=1}^m{a}_i\left(t\right)B_{i}y,y\left(0\right)=y_0\in R^m.}$$

(6)

Here, the ${\displaystyle (m\times m)}$ constant matrices ${\displaystyle A}$ and ${\displaystyle B_i,i=\overline{1,m}}$, are defined by

$${\displaystyle A=\left[0e_1\cdots e_{m-1}\right],B_1=\left[e_{1}0\cdots 0\right],\cdots ,B_m=\left[00\cdots e_m\right],}$$

(7)

where ${\displaystyle \left\{e_1,\dots ,e_m\right\}}$ is the canonical basis and ${\displaystyle 0\in R^m}$ is the origin. By definition

$${\displaystyle \left[B_i,B_j\right]:=B_j{B}_i-B_i{B}_j,i,j\in \left\{1,2,\dots ,m\right\},\left(Liebracket\right),}$$

(8)

and making a direct computation, we get

$${\displaystyle O=\left[B_i,B_j\right],i,j\in \left\{1,2,\dots ,m\right\},}$$

(9)

with ${\displaystyle O}$ - null matrix, and

$${\displaystyle A^m=O,\left(Aisanilpotentmatrix\right).}$$

(10)

The Cauchy Problem solution for $\left(4\right)$ is represented by

$${\displaystyle y(t;y_0)=\left[expAt\right]\widehat{y}(t;y_0),t\in I,}$$

(11)

where ${\displaystyle \left\{\widehat{y}(t;y_0):t\in I\right\}}$ fulfils the following linear system (initial value problem)

$${\displaystyle \frac{dy}{dt}=\sum _{i=1}^m{a}_i\left(t\right)A_i\left(t\right)y,t\in I,y\left(0\right)=y_0\in R^m.}$$

(12)

By using the linear mapping ${\displaystyle {ad}_A:M_{m\times m}\to M_{m\times m},{ad}_A\left(B\right):=BA-AB}$ (see ${\displaystyle \left[A,B\right]}$), write the ${\displaystyle (m\times m)}$ matrices

$${\displaystyle A_i\left(t\right):=\left[exp(-tA)\right]B_i\left[exptA\right],i\in \left\{1,2,\dots ,m\right\},}$$

(13)

as follows

$${\displaystyle A_i\left(t\right)=B_i+\frac{t}{1!}{ad}_A\left(B_i\right)+\cdots +\frac{t^{m-1}}{(m-1)!}{ad}_A^{m-1}\left(B_i\right),i\in \left\{1,\dots ,m\right\}.}$$

(14)

In addition, taking into account $\left(9\right)$, $\left(10\right)$ and $\left(14\right)$, we get

$${\displaystyle A_i\left(t\right)=\left[expt{ad}_A\right]\left(B_i\right),i\in \left\{1,2,\dots ,m\right\},t\in I.}$$

(15)

Denote ${\displaystyle N=m^2}$ and define ${\displaystyle N}$ matrices ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}\subseteq M_{m\times m}}$, as follows

$${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}=\left\{{ad}_A^k\left(B_1\right):k\in \left\{0,1,2,\dots ,m-1\right\}\right\}\cup \cdots \cup }$$

(16)

$${\displaystyle \left\{{ad}_A^k\left(B_m\right):k\in \left\{0,1,2,\dots ,m-1\right\}\right\}.}$$

Moreover, let $\left\{{\alpha }_1\left(t\right),{\alpha }_2\left(t\right),\dots ,{\alpha }_N\left(t\right):t\in I\right\}$ be given by

$${\displaystyle \left\{{\alpha }_1\left(t\right),\dots ,{\alpha }_m\left(t\right)\right\}=a_1\left(t\right)\left[1,\frac{t}{1!},\dots ,\frac{t^{m-1}}{(m-1)!}\right],}$$

(17)

$${\displaystyle ⋮}$$

$${\displaystyle \left\{{\alpha }_{N-m+1}\left(t\right),\dots ,{\alpha }_N\left(t\right)\right\}=a_m\left(t\right)\left[1,\frac{t}{1!},\dots ,\frac{t^{m-1}}{(m-1)!}\right].}$$

With these notations, we write ODE $\left(12\right)$ as follows

$${\displaystyle \frac{dy}{dt}=\sum _{j=1}^N{\alpha }_j\left(t\right)Y_j\left(y\right),t\in I,y\left(0\right)=y_0,}$$

(18)

where ${\displaystyle Y_j\left(y\right):=C_{j}y,j\in \left\{1,2,\dots ,N\right\}}$.

2. Main Results

In this section, the main results of the present paper are formulated and proved.

Theorem 1.

Consider ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ defined in $\left(16\right)$, with ${\displaystyle N=m^2}$. Then, ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ is a basis for ${\displaystyle M_{m\times m}}$ and

$${\displaystyle \left\{Y_1\left(y\right)=C_{1}y,Y_2\left(y\right)=C_{2}y,\dots ,Y_N\left(y\right)=C_{N}y\right\}}$$

is a system of generators for the Lie algebra ${\displaystyle L\left(Y_1,\dots ,Y_N\right)\subseteq C^{\infty }\left(R^m;R^m\right)}$ generated by ${\displaystyle \left\{Y_1,\dots ,Y_N\right\}}$.

Proof. 

By direct computation, we rewrite the matrices ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ as follows:

$${\displaystyle {ad}_A^k\left(B_1\right)=\left[{(-1)}^k{e}_{m-k}{c}_{1,1}^k{e}_{m-k+1}\dots c_{1,k}^k{e}_{m}0\dots 0\right],}$$

(19)

for some constants ${\displaystyle c_{1,j}^k,0\le k\le m-1}$. For any ${\displaystyle 1\le i\le m}$, we get

$${\displaystyle {ad}_A^k\left(B_i\right)=\left[\underset{(i-1)times}{\underbrace{0\dots 0}}{(-1)}^k{e}_{m-k}{c}_{i,1}^k{e}_{m-k+1}\dots c_{i,k\left(i\right)}^k{e}_{m-k+k\left(i\right)}0\dots 0\right],}$$

(20)

where ${\displaystyle k\left(i\right):=min\left\{k,m-i\right\}}$ and ${\displaystyle c_{i,j}^k}$ are some constants.

The particular structure given in $\left(20\right)$ leads us directly to the conclusion that ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}\subseteq M_{m\times m}}$ are linearly independent. Therefore, ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ is a basis for ${\displaystyle L\left(C_1,C_2,\dots ,C_N\right)=M_{m\times m}}$ and using

$${\displaystyle L\left(Y_1,Y_2,\dots ,Y_N\right)=\left\{Cy:C\in L\left(C_1,C_2,\dots ,C_N\right)\right\}}$$

(21)

we get that ${\displaystyle \left\{Y_1,Y_2,\dots ,Y_N\right\}}$ is a system of generators for ${\displaystyle L\left(Y_1,Y_2,\dots ,Y_N\right)}$. The proof is complete. □

The next remark contains several mathematical tools (some of these, introduced in Vârsan [1]) and their hints, which are necessary for proving Theorem 2.

Remark 1.

Consider the linear vector fields ${\displaystyle Y_j\left(y\right):=C_{j}y,j\in \left\{1,\dots ,N\right\}}$, ${\displaystyle y\in R^m}$, where ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}\subseteq M_{m\times m}}$ is a basis (see Theorem 1). Let ${\displaystyle L\left(Y_1,\dots ,Y_N\right)}$ be the finite dimensional Lie algebra generated by ${\displaystyle \left\{Y_1,\dots ,Y_N\right\}}$. Then, the following statements are valid:

$${\displaystyle dimL\left(Y_1,\dots ,Y_N\right)\left(y_0\right)=dim\left[span\left\{Y_1\left(y_0\right),\dots ,Y_N\left(y_0\right)\right\}\right]=m,}$$

(22)

for any ${\displaystyle y_0\ne 0\in R^m}$;

$${\displaystyle y(p;\lambda )=G\left(p\right)\lambda :=\left[exp{t}_1{C}_1\right]\cdots \left[exp{t}_N{C}_N\right]\lambda ,}$$

(23)

with ${\displaystyle p=(t_1,\dots ,t_N)\in R^N,\lambda \in R^m}$, satisfies a gradient system (GS) in ${\displaystyle L\left(Y_1,Y_2,\dots ,Y_N\right)}$ (see Vârsan [1] for more details; ${\displaystyle G\left(p\right)\in M_{m\times m},p\in R^N}$, is solution for a (GS) in ${\displaystyle M_{m\times m}}$)

$${\displaystyle {\partial }_{p}y(p;\lambda )=\left\{Y_1,\dots ,Y_N\right\}\left(y(p;\lambda )\right)A\left(p\right),p\in R^N,}$$

(24)

where ${\displaystyle A\left(p\right)}$ is an ${\displaystyle (N\times N)}$ analytic matrix fulfilling ${\displaystyle A\left(0\right)=I_N}$ and ${\displaystyle detA\left(p\right)\ne 0}$ for ${\displaystyle p\in B(0,\rho )\subseteq R^N}$; if

$${\displaystyle \phi \left(y(p;y_0)\right)=const.\left(=\phi \left(y_0\right)\right),\forall p\in B(0,\rho )\subseteq R^N,}$$

(25)

for some ${\displaystyle y_0\ne 0\in R^m}$ and ${\displaystyle \phi \in C^1\left(D\supseteq B(y_0,a)\right)}$, where ${\displaystyle B(y_0,a)\subseteq \left\{y(p;y_0)\ne 0:p\in B(0,\rho )\subseteq R^N\right\}}$, then ${\displaystyle {\partial }_y\phi \left(y\right)=0\in R^m,\forall y\in B(y_0,a)}$.

The conclusion $\left(22\right)$ relies on Theorem 1 and we get

$${\displaystyle dimL\left(Y_1,\dots ,Y_N\right)\left(y_0\right)=dim\left[span\left\{Y_1\left(y_0\right),\dots ,Y_N\left(y_0\right)\right\}\right]}$$

$${\displaystyle =dim\left[span\left\{C_1{y}_0,\dots ,C_N{y}_0\right\}\right]=dim\left\{C{y}_0:C\in M_{m\times m}\right\}=m,\forall y_0\ne 0\in R^m.}$$

Using (23) and (24), we compute the following Lie derivatives

$${\displaystyle 0=<{\partial }_p\left[\phi \left(y(p;y_0)\right)\right],A^{-1}\left(p\right)e_j>=<{\partial }_y\phi \left(y(p;y_0)\right),{\partial }_{p}y(p;y_0)A^{-1}\left(p\right)e_j>}$$

(26)

$${\displaystyle =<{\partial }_y\phi \left(y(p;y_0)\right),Y_j\left(y(p;y_0)\right)>,j\in \left\{1,2,\dots ,N\right\},}$$

where ${\displaystyle \left\{y(p;y_0)\ne 0:p\in B(0,\rho )\right\}\supseteq B(y_0,a)}$ and ${\displaystyle \left\{e_1,e_2,\dots ,e_N\right\}\subseteq R^N}$ is the canonical basis. Taking into account $\left(22\right)$ and $\left(26\right)$, we obtain ${\displaystyle {\partial }_y\phi \left(y(p;y_0)\right)=0,p\in B(0,\rho )}$, and ${\displaystyle {\partial }_y\phi \left(y\right)=0}$ for any ${\displaystyle y\in B(y_0,a)}$. In other words, as far as

$${\displaystyle dim\left[span\left\{Y_1\left(y_0\right),\dots ,Y_N\left(y_0\right)\right\}\right]=m,}$$

for any ${\displaystyle y_0\ne 0}$ (see $\left(22\right)$), from $\left(26\right)$, we obtain both ${\displaystyle {\partial }_y\phi \left(y(p;y_0)\right)=0\in R^m}$ for any ${\displaystyle p\in B(0,\rho )\subseteq R^N}$ and ${\displaystyle {\partial }_y\phi \left(y\right)=0\in R^m}$, for any ${\displaystyle y\in B(y_0,a)\subseteq R^m}$.

Theorem 2.

Assume ${\displaystyle a_j\left(t\right)\equiv 0,j\in \left\{1,2,\dots ,i\right\}}$, for some ${\displaystyle 1\le i\le m}$. Define a subspace ${\displaystyle S_i=span\left\{e_1,\dots ,e_i\right\}\subseteq R^m}$ and its orthogonal complement ${\displaystyle S_i^{\perp }\subseteq R^m}$, where ${\displaystyle \left\{e_1,\dots ,e_m\right\}\subseteq R^m}$ is the canonical basis. Then

$${\displaystyle \forall y_0\in S_{i}isastationarypointforODE\left(12\right).}$$

(27)

In addition, for each ${\displaystyle y_0\in S_i^{\perp },y_0\ne 0}$, the following statements are valid:

$${\displaystyle dimL\left(Y_{j\left(i\right)},\dots ,Y_N\right)\left(y_0\right)=dim\left[span\left\{Y_{j\left(i\right)}\left(y_0\right),\dots ,Y_N\left(y_0\right)\right\}\right]=m-i,}$$

(28)

$${\displaystyle \left\{C_{j\left(i\right)},\dots ,C_N\right\}isabasisfor{M}_{m\times m}^i:=L\left(C_{j\left(i\right)},\dots ,C_N\right),j\left(i\right):=mi+1}$$

$${\displaystyle \widehat{y}(t,y_0)\in S_i^{\perp },t\in I,foranysolutionofODE\left(12\right).}$$

(29)

Proof. 

As far as any ${\displaystyle y_0\in S_i}$ is a stationary point for each ${\displaystyle Y_j\left(y\right)=C_{j}y,j=j\left(i\right),\dots ,N,j\left(i\right)=mi+1}$, and ODE $\left(12\right)$ is written using only ${\displaystyle \left\{Y_{j\left(i\right)}\left(y\right),\dots ,Y_N\left(y\right)\right\}}$, we get the conclusion $\left(27\right)$ is satisfied. By definition, (see Theorem 1) the matrices ${\displaystyle \left\{C_{j\left(i\right)},\dots ,C_N\right\}\subseteq M_{m\times m}^i}$ determine a basis in the space ${\displaystyle M_{m\times m}^i}$ consisting of all ${\displaystyle \left(m\times m\right)}$ matrices ${\displaystyle C=\left[\underset{i-times}{\underbrace{0\dots 0}}{c}_{i+1}\dots c_m\right]}$, with ${\displaystyle c_j\in R^m,j=i+1,\dots ,m}$. Using

$${\displaystyle dimL\left(Y_{j\left(i\right)},\dots ,Y_N\right)\left(y_0\right)=dim\left[span\left\{Y_{j\left(i\right)}\left(y_0\right),\dots ,Y_N\left(y_0\right)\right\}\right]}$$

$${\displaystyle =dim\left[span\left\{C_{j\left(i\right)}{y}_0,\dots ,C_N{y}_0\right\}\right]}$$

$${\displaystyle =dim\left\{C^i{y}_0:C^i\in M_{m\times m}^i\right\}=m-i,}$$

we get the conclusion $\left(28\right)$. Notice that ${\displaystyle \frac{d}{dt}<\widehat{y}(t;y_0),e_j>=0,t\in I}$, and ${\displaystyle <\widehat{y}(t;y_0),e_j>=<y_0,e_j>=0,t\in I}$, for any ${\displaystyle j=\overline{1,i}}$, which stands for the conclusion $\left(29\right)$. The proof is complete. □

3. Conclusions

In this paper, we investigated (by using the characteristic system method) the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. In addition, a gradient representation for the associated Cauchy Problem solution has been formulated. Moreover, as further research directions, some applications were highlighted in this study of some higher-order hyperbolic, parabolic or Hamilton-Jacobi equations involving a finite set of derivations.