Erratum to: Semigroups of max-plus linear operators

1 Erratum to: Semigroup Forum DOI 10.1007/s00233-015-9761-x

Boris Andreianov determined that the examples of max-additive and max-plus linear semigroups in the last section of the article [6] are given inaccurately, i.e., [6, Proposition 4.1] is not true as stated and [6, Proposition 4.2] does not hold without some additional assumptions. Jointly we are able to correct the issues as follows.

In the case of conservation law (CL) studied in [6, Subsec. 4.1], a smaller set of weak solution should be considered. A function $v:[0,\infty )\rightarrow \mathrm {L}^1(\mathbb {R})$ is called an isentropic solution to (CL), if it satisfies the initial condition $v(0)=h$ and the Kružkov condition (14) from [6] with the equality sign.

We now give the proper formulation of [6, Proposition 4.1] and its proof. Let

$$\begin{aligned} \begin{aligned} \mathcal {C}:=&\{h\in \mathrm {L}^1(\mathbb {R})\mid (CL) \text { admits an isentropic solution } v \text { with }v(0)=h\}\\&\cup \{-\infty \}. \end{aligned} \end{aligned}$$

Proposition

([6, Proposition 4.1]) The semigroup $\mathcal {T}^{\mathrm {CL}}:=\left( T(t)\right) _{t\ge 0}$, where $u(t,x)=T(t)h(x)$ is the unique isentropic solution to (CL) and where $T(t)(-\infty ):= -\infty$, is a max-additive strongly continuous semigroup on $\mathcal {C}$.

Proof

By [5, Theorem 1], $T(t)h_1\oplus T(t)h_2$ is an isentropic solution to (CL) for any $h_1,h_2\in \mathcal {C}$. Moreover, $T(t) h_1\oplus T(t) h_2\vert _{t=0} = h_1\oplus h_2$, hence $T(t) h_1\oplus T(t) h_2$ is the isentropic solution to (CL) with initial condition $h=h_1\oplus h_2\in \mathcal {C}$. By uniqueness of the isentropic solutions, we obtain $T(t)h_1\oplus T(t) h_2 = T(t) (h_1\oplus h_2)$.$\square$

Remark 1

The isentropic solutions to (CL) often exist only on some limited time interval. However, our results also hold in this case considering the local time flows $\left( T(t)\right) _{t\in [0,T]}$ defined on

$$\begin{aligned} \begin{aligned} \mathcal {C}_T:=\{&h\in \mathrm {L}^1(\mathbb {R})\mid (CL) \text { admits an isentropic solution } v \text { on } [0,T] \\&\text { with }v(0)=h\} \cup \{-\infty \}. \end{aligned} \end{aligned}$$

In the case of Hamilton–Jacobi equation (HJ) considered in [6, Subsec. 4.2], we should specify the choice of the generalized solutions and regularity assumptions. We assume the following for the function f appearing in the problem (HJ).

• 1. For every $x\in \mathbb {R}^n$, $\Vert x\Vert =1$, there exists the limit $\lim _{r\downarrow 0} r f(x/r)$.

• 2. For any $(x,r), (x',r')\in \mathbb {R}^n\times \mathbb {R}_+$ with $\Vert x\Vert ^2 +r^2 \le 1$ and $\Vert x'\Vert ^2 +r'^2 \le 1$,

  $$\begin{aligned} \left| r f(x/r)-r' f(x'/r')\right| \le K \left( \Vert x-x'\Vert ^2 + (r-r')^2\right) ^{1/2}, \end{aligned}$$

  where $K>0$ is some constant.

• 3. The function f is convex.

In the literature, there are many notions for generalized weak solutions to (HJ): Crandall–Lions’ viscosity solutions, minimax solutions by Subbotin, Maslov idempotent weak solutions, Kružkov generalized solutions. However, under our assumptions they all agree, see [7, Sec. 5], and [6, Proposition 4.2] holds as stated in the paper. For the proof, we refer directly to [7, Theorem 3.2] (note that there the min-plus terminology is used; therefore, the concavity instead of convexity of f is assumed).

Remark 2

Stability of solutions of (HJ) (understood in the viscosity sense of Crandall–Lions) under the $\oplus$-operation is closely related to the properties of $\mathrm {liminf}$ of a sequence of viscosity solutions of first order Hamilton–Jacobi equations with convex Hamiltonians, see [2] and [1, Th. 2.1 and 2.3]. The fact that the maximum of two viscosity solutions is a viscosity sub-solution is classical, see [3]. Heuristically, the fact that the maximum of two viscosity solutions is also a viscosity super-solution stems from the semi-concavity property of viscosity solutions, see [1, p.1125].

Remark 3

We add a reference for yet another proof of the max-linearity of Hamilton–Jacobi–Bellman semigroup presented in Subsection 4.3. In [4], this fact is proved using a probabilistic approach.