Local Null Controllability of a 1D Stefan Problem

Abstract

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

Appendix: Some Technical Results

1.1 Differentiability of $M^{\ell }$

Now we proceed to study the applications which make correspond to any $\ell \in C^1([0,T])$ the coefficients of the differential operator $M^{\ell }$, namely, $\ell \mapsto a^{\ell }$ and $\ell \mapsto b^{\ell }$. From now on, we will use the notations $a^{\ell }$ and $a(\ell )$ indistinctly.

Let us define $da(\ell ,\tilde{\ell }) (\xi ,t) := \frac{d}{d \epsilon } a^{\ell + \epsilon \tilde{\ell }}(\xi ,t) \Big |_{\epsilon =0}.$ Using the chain rule we can see that $da(\ell ,\tilde{\ell }) (\xi ,t)$ is equal to

$$\begin{aligned}&H_y\big ( x(\xi ,t) , \ell (t) \big ) \tilde{\ell }'(t) + \nabla H_y \big ( x(\xi ,t) , \ell (t) \big ) \cdot \Big ( H_y\big ( x(\xi ,t) , \ell (t) \big ) , 1 \Big ) \ell '(t) \tilde{\ell }(t) \nonumber \\&\quad +\, \nabla H_{xx} \big (x(\xi ,t) , \ell (t) \big ) \cdot \Big ( H_y\big ( x(\xi ,t) , \ell (t) \big ) , 1 \Big ) \tilde{\ell }(t), \end{aligned}$$

(6.1)

where $x(\xi ,t)$ denotes $H^{-1}(\xi ,\ell (t))$. Analogously, we obtain

$$\begin{aligned}&db(\ell ,\tilde{\ell }) (\xi ,t) \nonumber \\&\quad =\, 2 H_{x} \big ( x(\xi ,t) , \ell (t) \big ) \nabla H_{x} \big ( x(\xi ,t) , \ell (t) \big ) \cdot \Big ( H_y\big ( x(\xi ,t) , \ell (t) \big ) , 1 \Big ) \tilde{\ell }(t). \end{aligned}$$

(6.2)

Now we proceed to verify that $da(\ell , \tilde{\ell })$ (resp. $da_{\xi }(\ell , \tilde{\ell })$ and $db(\ell , \tilde{\ell })$) is in fact the Gâteaux-derivative of a (resp. b) at $\ell$ along the direction $\tilde{\ell }$. Here, we deal with $b^{\ell } = H^2_{x}\big ( H^{-1}(\xi ,\ell (t)) , \ell (t) \big )$. The other cases can be handled in a similar way.

Let us fix $x=H^{-1}(\xi ,\ell (t))$, $y= \ell (t)$, $h_1=H^{-1}(\xi ,\ell (t)+ \epsilon \tilde{\ell }(t)) - H^{-1}(\xi ,\ell (t))$ and $h_2= \epsilon \tilde{\ell }(t)$. Using the Taylor’s first order expansion of $H^2_x$, we obtain:

$$\begin{aligned} \Big | b^{\ell + \epsilon \tilde{\ell }}(\xi ,t) -b^\ell (\xi ,t) - \nabla H^2_x(x,y) \cdot (h_1,h_2) \Big | \le C_1 \Vert (h_1,h_2) \Vert ^2 , \end{aligned}$$

where $C_1$ can be taken as the supremum of $\Vert \text {Hess}(H^2_x)\Vert$ on a fixed domain, says $\overline{Q_{\ell +|\tilde{\ell }|}}$. On the other hand

$$\begin{aligned} \Big | H^{-1}(\xi ,\ell (t)+ \epsilon \tilde{\ell }(t)) - H^{-1}(\xi ,\ell (t)) -\nabla H^{-1}(\xi ,\ell (t)) \cdot (0,\epsilon \tilde{\ell }(t)) \Big | \le C_2 |\epsilon \tilde{\ell }(t)|^2 \end{aligned}$$

where $C_2:=\sup _{(z,w)\in H^{-1}(\overline{Q_{\ell +|\tilde{\ell }|}})} \Vert \text {Hess}(H^{-1} (z,w))\Vert$. Therefore,

$$\begin{aligned}&\Big | b^{\ell + \epsilon \tilde{\ell }}(\xi ,t) -b^\ell (\xi ,t) - \epsilon db(\ell ,\tilde{\ell }) \Big |\\&\quad \le \Big | b^{\ell + \epsilon \tilde{\ell }}(\xi ,t) -b^\ell (\xi ,t) - \nabla H^2_x(x,y) \cdot (h_1,h_2) \Big |\\&\qquad +\Big | \nabla H^2_x(x,y) \cdot (h_1- \epsilon H_y\big ( x , y \big ) \tilde{\ell }(t) \; , \;0) \Big | \end{aligned}$$

and thus

$$\begin{aligned} \Big | \frac{b^{\ell + \epsilon \tilde{\ell }}(\xi ,t) -b^\ell (\xi ,t)}{\epsilon } - db(\ell ,\tilde{\ell }) \Big |&\le 3 C \epsilon |\tilde{\ell }(t)|^2, \end{aligned}$$

where we can take

$$\begin{aligned} C= C_1 C_2 \sup _{(x,t) \in \overline{Q_{\ell +|\tilde{\ell }|}}} \Vert \nabla H^2_x(x,t) \Vert . \end{aligned}$$

1.2 Hölder Regularity of Parabolic Equations

Given a connected open set $\Omega \subset \mathbb R^n$, consider the cylinder $Q_T= \Omega \times (0,T)$ and denote by $S_T=\{(x,t) : x \in \partial \Omega , \ t \in [0,T]\}$ its lateral surface. Let $\mathcal C^{\alpha ,\alpha /2}(\overline{Q_T})$ be, for any $\alpha \in (0,1]$, the space of functions $u: \overline{Q_T} \mapsto \mathbb R$ such that $D^r_t D^s_x u$ is:

• (i) Continuous on $\overline{\Omega }$ for $0 \le 2r + s \le \lfloor \alpha \rfloor$.

• (ii) $\gamma$-Hölder continuous in space of index $\gamma = \alpha - \lfloor \alpha \rfloor$ for $2r + s = \lfloor \alpha \rfloor$.

• (iii) $\gamma$-Hölder continuous in time of index $\gamma = (\alpha - 2r - s)/2$ for $0< \alpha - 2r - s < 2$.

Obviously, $\mathcal C^{\alpha ,\alpha /2}(\overline{Q_T})$ is a Banach space for the norm

$$\begin{aligned} \Vert \cdot \Vert _{ \mathcal C^{\alpha ,\alpha /2}(\overline{Q_T})}&= \sum _{0 \le 2r + s \le \lfloor \alpha \rfloor } \Vert D^r_t D^s_x u \Vert _{\infty } \, + \, \sum _{2r + s = \lfloor \alpha \rfloor } \Vert D^r_t D^s_x u \Vert _{C^{\alpha - \lfloor \alpha \rfloor }_{x}(\overline{Q_T})} \\&\quad \, + \, \sum _{0< \alpha - 2r - s < 2 } \Vert D^r_t D^s_x u \Vert _{C^{(\alpha - 2r - s)/2}_{x}(\overline{Q_T})}, \end{aligned}$$

where $\Vert \cdot \Vert _{\infty }$ denotes the norm of the uniform convergence and

$$\begin{aligned} \Vert u\Vert _{C_x^{\gamma }(\overline{Q_T} )}= & {} \sup _{(x,t),(y,t) \in \overline{Q_T} } \frac{|u(x,t) - u(y,t)|}{ |x-y|^{\gamma }}, \\ \Vert u\Vert _{C_t^{\gamma }(\overline{Q_T} )}= & {} \sup _{(x,s),(x,t) \in \overline{Q_T} } \frac{|u(x,s) - u(x,t)|}{|s-t|^{\gamma }}. \end{aligned}$$

Consider the differential operator

$$\begin{aligned} \mathcal L u = u_t - \sum _{i,j=1}^n a_{i,j}(x,t ) \ u_{x_i,x_j} + \sum _{i=1}^n a_{i}(x,t ) \ u_{x_i} + a(x,t)u . \end{aligned}$$

We will say that the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal L u = f(x,t),\\ u \big |_{t=0} = \phi (x),\\ u \big |_{S_T} = \Phi (x,t). \end{array}\right. } \end{aligned}$$

(6.3)

satisfies the compatibility conditions of order $m\ge 0$ if

$$\begin{aligned} u^{(k)}(x) \big |_{x \in S} = \frac{\partial ^{k} u}{\partial t^k}\Big |_{t=0} = \frac{\partial ^{k} \Phi }{\partial t^k}\Big |_{t=0} = \Phi ^{(k)}(x) \ \ \text {for} \ \ k=0, \cdots , m. \end{aligned}$$

The next two results are classical and well known (see Theorems III.12.2 and IV.5.2 of Ladyzenskaja et al. 1968):

Theorem 6.1

Suppose $u \in W_q^{2,1}(Q_T)$ is a generalized solution of $\mathcal L u = f(x,t)$. If f and the coefficients of the operator $\mathcal L$ belong to $\mathcal C^{\alpha ,{\alpha }/{2}} (Q_T)$, then u belongs to $\mathcal C^{2+\alpha ,1+\alpha /2} (Q_T)$.

See also the comments in Ladyzenskaja et al. (1968, p. 223), below the statement of Theorem III.12.2.

Theorem 6.2

Suppose that $\alpha >0$, the coefficients of the operator $\mathcal L$ belong to $\mathcal C^{\alpha ,\alpha /2}(\overline{Q_T})$ and the boundary S is sufficiently regular (more precisely, of class $\mathcal C^{\alpha + 2}$). Then, for any $f \in \mathcal C^{\alpha ,\alpha /2}(\overline{Q_T})$, $\phi \in \mathcal C^{\alpha +2}(\overline{\Omega })$ and $\Phi \in \mathcal C^{\alpha + 2,{\alpha }/{2} + 1}(\overline{S_T})$ satisfying the compatibility conditions of order $\lceil \alpha /2\rceil$, (6.3) possesses exactly one solution in $\mathcal C^{\alpha + 2,\alpha /2 + 1}(\overline{Q_T})$.