Stability for evolution equations governed by a non-autonomous form

TL;DR

This paper investigates the stability and convergence of solutions to non-autonomous evolution equations governed by time-dependent forms, establishing conditions under which maximal regularity is preserved in the limit.

Contribution

It demonstrates that the limit of a sequence of non-autonomous forms with maximal regularity also retains maximal regularity under certain assumptions.

Findings

Maximal regularity is preserved in the limit for converging sequences of non-autonomous forms.

Convergence of solutions is uniform with respect to initial data and inhomogeneity.

The paper provides conditions ensuring stability of solutions under form approximation.

Abstract

This paper deals with the approximation of non-autonomous evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation*} where $A(t),\ t\in [0,T]$ arise from a non-autonomous sesquilinear forms $\mathfrak a(t;\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ Assuming the existence of a sequence $\mathfrak a_n:[0,T]\times V\times V\longrightarrow\mathbb C, n\in \mathbb N$ of non-autonomous forms such that the associated Cauchy problem has $L^2$-maximal regularity in $H$ and $\mathfrak a_n(t,u,v)$ converges to $\mathfrak a(t,u,v)$ as $n\to \infty,$ then among others we show under additional assumptions that the limit problem has $L^2$-maximal regularity. Further we show that the convergence is uniformly on the initial data $u_0$ and the inhomogeneity $f.$

Equations

$$\dot{u}(t)+\mathcal{A}(t)u(t)=f(t)t\in [0,T],u(0)=u_0.$$

$$|\mathfrak{a}(t;u,v)|\le M\Vert u{\Vert }_V\Vert v{\Vert }_V,\text{ and }\mathrm{Re}\text{leavevmode}\mathfrak{a}(t;u,u)+\beta \Vert u{\Vert }_H^2\ge \alpha \Vert u{\Vert }_V^2$$

$$⟨\mathcal{A}(t)u,v⟩=\mathfrak{a}(t;u,v),u,v\in V.$$

$$D(A(t)):=$$

$$A(t)u:=$$

$$\dot{u}(t)+\mathcal{A}(t)u(t)=f(t),a.e\text{o}n[0,T],u(0)=u_0.$$

$$\|u\|_{MR_{2}(V,V^{\prime})}\leq c_{0}\big{[}\|f\|_{L^{2}(0,T;V^{\prime})}+\|u_{0}\|_{H},\big{]}$$

$${\dot{u}}_n(t)+{\mathcal{A}}_n(t)u_n(t)=f(t),u(0)=u_0$$

$$|{\mathfrak{a}}_n(t,u,v)-\mathfrak{a}(t,u,v)|\le d_n\Vert u{\Vert }_V\Vert v{\Vert }_{V_{\gamma }},t\in [0,T],u,v\in V,$$

$$|{\mathfrak{a}}_n(t,u,v)-{\mathfrak{a}}_n(s,u,v)|\le {\omega }_n(|t-s|)\Vert u{\Vert }_V\Vert v{\Vert }_{V_{\gamma }},t\in [0,T],u,v\in V,$$

$$\displaystyle\displaystyle\sup_{t\in[0,T],n\in\mathbb{N}}\frac{\omega_{n}(t)}{t^{\gamma/2}}<\infty,\text{ and }\ \ \displaystyle\sup_{n\in\mathbb{N}}\int_{0}^{T}\frac{\omega_{n}(t)}{t^{1+\gamma/2}}{\rm d}t<\infty\ \$$

$${\dot{u}}_n(t)+{\mathcal{A}}_n(t)u_n(t)=f(t),a.e\text{o}n[0,T],u_n(0)=u_0,(n\in \mathbb{N}).$$

$$H=Tr_{2}(V,V^{\prime}):=\Big{\{}u(0)\,|\,u\in MR_{2}(V,V^{\prime})\Big{\}}.$$

$$\|u_{n}-u\|_{MR_{2}(V,V^{\prime})}\leq\mathbf{c}d_{n}\Big{[}\|f\|_{L^{2}(0,T;V^{\prime})}+\|u_{0}\|_{H}\Big{]},\ n\in\mathbb{N},\qquad\text{ and}$$

$$\|u_{n}-u\|_{C([0,T],H)}\leq\mathbf{c}d_{n}\Big{[}\|f\|_{L^{2}(0,T;V^{\prime})}+\|u_{0}\|_{H}\Big{]},\ n\in\mathbb{N}$$

$$(\mathcal{A}f)(t)={\mathcal{A}}_n(t)f(t),({\mathcal{A}}_{n}f)(t)={\mathcal{A}}_n(t)f(t)and(\mathcal{B}u)(t)=\dot{u}(t)for\; almost\; everyt\in [0,T].$$

$$D(\mathcal{A}+\mathcal{B})=D({\mathcal{A}}_n+\mathcal{B}):=\{u\in M{R}_2(V,V^{\prime })|u(0)=0\}$$

$$\Vert u_n-u{\Vert }_{M{R}_2(V,V^{\prime })}$$

$$=\Vert (\mathcal{A}+\mathcal{B}{)}^{-1}({\mathcal{A}}_n-\mathcal{A})({\mathcal{A}}_n+\mathcal{B}{)}^{-1}f{\Vert }_{M{R}_2(V,V^{\prime })}$$

$$\le \mathbf{c}{d}_n\Vert f{\Vert }_{L^2(0,T;V^{\prime })}.$$

$$\dot{v}_{n}(t)+\mathcal{A}_{n}(t)v_{n}(t)=g_{n}(t)\quad{a.e}\quad\text{o}n\quad[0,T],\ \ \ \ \ v_{n}(0)=0,\$$

$$\dot{v}(t)+\mathcal{A}(t)v(t)=g(t)\quad{a.e}\quad\text{o}n\quad[0,T],\ \ \ \ \ v(0)=0.\$$

$$\Vert u_n-u{\Vert }_{M{R}_2(V,V^{\prime })}=$$

$$\le \Vert ({\mathcal{A}}_n+\mathcal{B}{)}^{-1}(g_n-g){\Vert }_{M{R}_2(V,V^{\prime })}+\Vert ({\mathcal{A}}_n+\mathcal{B}{)}^{-1}g-(\mathcal{A}+\mathcal{B}{)}^{-1}g{\Vert }_{M{R}_2(V,V^{\prime })}$$

$$\le \mathbf{c}\Vert {\mathcal{A}}_n\vartheta -\mathcal{A}\vartheta {\Vert }_{L^2(0,T;V^{\prime })}+\mathbf{c}{d}_n\Vert g{\Vert }_{L^2(0,T;V^{\prime })}$$

$$\displaystyle\leq\mathbf{c}d_{n}\Big{[}\|\vartheta\|_{L^{2}(0,T;V)}+\|\dot{\vartheta}\|_{L^{2}(0,T;V^{\prime})}+\|f\|_{L^{2}(0,T;V^{\prime})}\Big{]}$$

$$\displaystyle\leq\mathbf{c}d_{n}\Big{[}\|u_{0}\|_{H}+\|f\|_{L^{2}(0,T;V^{\prime})}\Big{]}.$$

$$({\mathbf{H}}_1)|{\mathfrak{a}}_n(t,u,v)-\mathfrak{a}(t,u,v)|\le d_n\Vert u{\Vert }_V\Vert v{\Vert }_{V_{\gamma }},t\in [0,T],u,v\in V,$$

$$({\mathbf{H}}_2)|{\mathfrak{a}}_n(t,u,v)-{\mathfrak{a}}_n(s,u,v)|\le {\omega }_n(|t-s|)\Vert u{\Vert }_V\Vert v{\Vert }_{V_{\gamma }},t\in [0,T],u,v\in V,$$

$$\displaystyle\mathbf{(H_{3})}\quad\displaystyle\sup_{t\in[0,T],n\in\mathbb{N}}\frac{\omega_{n}(t)}{t^{\gamma/2}}<\infty,\text{ and }\ \ \displaystyle\sup_{n\in\mathbb{N}}\int_{0}^{T}\frac{\omega_{n}(t)}{t^{1+\gamma/2}}{\rm d}t<\infty\ \$$

$$({\mathbf{H}}_4)\text{The approximation problem (refApproximation Abstract Cauchy problem) has L2-maximal regularity in H for every }n\in \mathbb{N}.$$

$$V\hookrightarrow V_{\gamma }\hookrightarrow H\hookrightarrow V_{\gamma }^{\prime }\hookrightarrow V^{\prime }$$

$$\Vert {\mathcal{A}}_n(t)-{\mathcal{A}}_n(s){\Vert }_{\mathcal{L}(V,V_{\gamma }^{\prime })}\le {\omega }_n(|t-s|),t,s\in [0,T],n\in \mathbb{N}.$$

$$u_n(t)=e^{-t{A}_n(t)}{u}_0+{\int }_0^t{e}^{-(t-s){\mathcal{A}}_n(t)}f(s)\mathrm{d}s+{\int }_0^t{e}^{-(t-s)A_n(t)}({\mathcal{A}}_n(t)-{\mathcal{A}}_n(s))u_n(s)\mathrm{d}s$$

$$u_{n,1}(t):=e^{-t{A}_n(t)}{u}_0,u_{n,2}(t):={\int }_0^t{e}^{-(t-s)A_n(t)}f(s)\mathrm{d}s.$$

$$(Q_n^{\mu }g)(t):={\int }_0^t({\mathcal{A}}_n(t)+\mu )e^{-(t-s)({\mathcal{A}}_n(t)+\mu )}({\mathcal{A}}_n(t)-{\mathcal{A}}_n(s))({\mathcal{A}}_n(s)+\mu {)}^{-1}g(s)\mathrm{d}st\text{-a.e}.$$

Full text

[TABLE]

where $\mathcal{A}(t),\ t\in[0,T]$ arise from a non-autonomous sesquilinear forms $\mathfrak{a}(t;\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ Assuming the existence of a sequence $\mathfrak{a}_{n}:[0,T]\times V\times V\longrightarrow\mathbb{C},n\in\mathbb{N}$ of non-autonomous forms such that the associated Cauchy problem has $L^{2}$-maximal regularity in $H$ and $a_{n}(t,u,v)$ converges to $a(t,u,v)$ as $n\to\infty,$ then among others we show under additional assumptions that the limit problem has $L^{2}$-maximal regularity. Further we show that the convergence is uniformly on the initial data $u_{0}$ and the inhomogeneity $f.$

Introduction

Throughout this paper $H,V$ are two separable Hilbert spaces over $\mathbb{K}=\mathbb{C}.$ We denote by $(\cdot\,|\,\cdot)_{V}$ the scalar product and $\|\cdot\|_{V}$ the norm on $V$ and by $(\cdot\,|\,\cdot),\|\cdot\|$ the corresponding quantities in $H.$ Moreover, we assume that $V$ is densely and continuously embedded into $H.$ Let $V^{\prime}$ denote the antidual of $V$ and $\langle\cdot,\cdot\rangle$ the duality between $V^{\prime}$ and $V$. As usual, by identifying $H$ with $H^{\prime},$ we have $V\hookrightarrow H\cong H^{\prime}\hookrightarrow V^{\prime}$ with continuous and dense embedding. Let $T>0.$ Let $\mathfrak{a}:[0,T]\times V\times V\longrightarrow C$ be a closed non-autonomous sesquilinear form, i.e., $\mathfrak{a}(.,u,v)$ is measurable for all $u,v\in V,$ and $\mathfrak{a}(t;\cdot,\cdot)$ is a sesquilinear form with

[TABLE]

for all $t\in[0,T],v,u\in V$ and for some constants $\beta\in\mathbb{R},\alpha,M>0.$ By Lax-Milgram Theorem, for each $t\in[0,T]$ there exists an isomorphism $\mathcal{A}(t):V\to V^{\prime}$ such that

[TABLE]

We call $\mathcal{A}(t)$ the operator associated with $\mathfrak{a}(t,\cdot,\cdot)$ on $V^{\prime}.$ Seen as an unbounded operator on $V^{\prime}$ with domain $D(\mathcal{A}(t))=V,$ the operator $-\mathcal{A}(t)$ generates a holomorphic $C_{0}-$semigroup $\mathcal{T}$ on $V^{\prime}.$ Further, we denote by $A(t)$ the part of $\mathcal{A}(t)$ on $H$; i.e.,

[TABLE]

It is a known fact that $-A(t)$ generates a holomorphic $C_{0}$-semigroup $T$ on $H$ and $T=\mathcal{T}_{\,|\,H}$ is the restriction of the semigroup generated by $-\mathcal{A}$ to $H.$ Then $A(t)$ is the operator induced by $\mathfrak{a}(t,.,.)$ on $H.$ See, e.g., [2],[15], [22, Chap. 2] and [8].

Consider the non-autonomous Cauchy problem

[TABLE]

Then the following $L^{2}$-maximal regularity in $V^{\prime}$ result has been proved by J. L. Lions on 1961:

Theorem 0.1**.**

(Lions 1961) The non-autonomous Cauchy problem (2) has $L^{2}$-maximal regularity in $V^{\prime},$ i.e., for given $f\in L^{2}(0,T;V^{\prime})$ and $u_{0}\in H,$ (2) has a unique solution $u$ in $MR_{2}(V,V^{\prime}):=L^{2}(0,T;V)\cap H^{1}(0,T;V^{\prime}).$ Moreover, there exists a constant $c_{0}>0$ depending only on $\alpha,\beta,M$ and $c_{H}$ such that

[TABLE]

where $c_{H}$ is the continuous embedding constant of $V$ into $H.$

Lions proved this result in [17] (see also [23, Chapter 3]) using a representation theorem of linear functionals due to him self and usually known in the literature as Lions’s representation Theorem and using Galerkin’s method in [9, XVIII Chapter 3, p. 620]. We refer [22, Section 5.5] and [21] for other proofs. The theorem of Lions requires only the measurability of $t\mapsto\mathfrak{a}(t;u,v)$ for all $u,v\in V$. However, in applications to boundary problems maximal regularity in $V^{\prime}$ is not sufficient because it is only the part $A(t)$ of $\mathcal{A}(t)$ in $H$ that realizes the boundary conditions in question. Precisely one is more interested on $L^{2}$-maximal regularity in $H,$ i.e., the solution $u$ of (2) belong to $H^{1}(0,T;H)$ if $f\in L^{2}(0,T;H)$ and $u_{0}\in V.$ The problem of $L^{2}$-maximal regularity in $H$ was initiated by Lions in [17, p. 68] for $u_{0}=0$ and $\mathfrak{a}$ is symmetric. In general, we have to impose more regularity on the form $\mathfrak{a}$ then measurability of the form is not sufficient [10, 3]. However, under additional regularity assumptions on the form $\mathfrak{a},$ the initial value $u_{0}$ and the inhomogeneity $f,$ some positive results were already done by Lions in [17, p. 68, p. 94, ], [17, Theorem 1.1, p. 129] and [17, Theorem 5.1, p. 138] and by Bardos [7]. More recently, this problem has been studied with some progress and different approaches [4, 5, 18, 11, 16, 19, 13, 12, 14]. Results on multiplicative perturbation are established in [4, 11, 6]. See also the recent review paper [3] for more details and references.

Exploiting the ideas and proofs of a recent result of Arendt and Monniaux [5] we study in this paper stability and the uniform approximation of the non-autonomous Cauchy problems (2). More precisely, assume that there exists a sequence $\mathfrak{a}_{n}:[0,T]\times V\times V\longrightarrow\mathbb{C}$ of non-autonomous forms such that Cauchy problem

[TABLE]

associated with $\mathfrak{a}_{n}$ has $L^{2}$-maximal regularity in $H$ and $a_{n}(t,u,v)$ converges to $a(t,u,v)$ as $n\to\infty.$ Then our aim is to study weather $L^{2}$-maximal regularity is iterated by the limit problem (2) and weather the sequence $(u_{n})_{n\in\mathbb{N}}$ of solutions of (4) converges uniformly on $u_{0}$ and $f$ to the the solution of (2). Let $0<\gamma<1.$ Let $\omega_{n}:[0,T]\longrightarrow[0,+\infty),n\in\mathbb{N},$ be a sequence of non-decreasing continuous function and let $(d_{n})_{n\in\mathbb{N}}$ be a zero real sequence such that

[TABLE]

for all $t,s\in[0,T],n\in\mathbb{N}$ and for all $u,v\in V,$ where $V_{\gamma}:=[H,V]_{\gamma}$ is the complex interpolation space. Then we show in Section 2 that the limit problem (2) has also $L^{2}$-maximal regularity in $H$ and the sequence $(u_{n})_{n\in\mathbb{N}}$ of solutions of (4) converges weakly in $MR_{2}(V,H)$ to the solution of (2). This convergences holds for the strongly topology of $MR_{2}(V,H)$ and uniformly on $u_{0}$ and $f$ provided the sequence $(d_{n})_{n\in\mathbb{N}}$ is decreasing, $\lim\limits_{n\to\infty}d_{n}n^{\gamma/2}=0$ and $\lim\limits_{n\to\infty}\displaystyle\int_{0}^{1/n}\frac{\omega_{n}(r)}{r^{{1+\gamma/2}}}{\rm d}r=0,$ see Section 3. Moreover we show that similar results holds on the space $C(0,T;V)$ if $(u_{n})_{n\in\mathbb{N}}\subset C(0,T;V).$ In the last section we provide an explicit approximation of $\mathfrak{a}$ that satisfies the above required hypothesis. The reader interested in examples of application is referred to above cited papers and the references therein.

1. Preliminary results: uniform approximation on $V^{\prime}$

In this section $\mathfrak{a}:[0,T]\times V\times V\longrightarrow C$ is a closed non-autonomous sesquilinear form. Moreover, we assume that there exist a sequence of closed non-autonomous sesquilinear forms $\mathfrak{a}_{n}:[0,T]\times V\times V\longrightarrow C$ satisfying (1) with the same constants $\beta,\alpha$ and $M>0$ and a zero real sequence $(d_{n})_{n\in\mathbb{N}}$ such that the following assumption holds:

${\bf(H_{0})}\ |\mathfrak{a}(t;u,v)-\mathfrak{a}_{n}(t;u,v)|\leq d_{n}\|u\|_{V}\|v\|_{V},\ t\in[0,T],u,v\in V.$

For each $t\in[0,T]$ and $n\in\mathbb{N},$ let $\mathcal{A}_{n}(t)\in\mathcal{L}(V,V^{\prime})$ be the operator associated with $\mathfrak{a}_{n}(t;\cdot,\cdot)$ on $V^{\prime}$ and consider the approximation Cauchy problems

[TABLE]

Note that the maximal regularity space $MR_{2}(V,V^{\prime})$ is continuously embedded into $C([0,T];H)$ [20, p. 106]. Moreover, the result of Lions implies that $H$ coincides with the trace space, that is

[TABLE]

The following theorem is the main results of this section.

Theorem 1.1**.**

Let $u,u_{n}\in MR_{2}(V,V^{\prime})$ be the solutions of (2) and (5), respectively. Then the following inequalities

[TABLE]

[TABLE]

hold for some positive constant $\mathbf{c}>0$ depending only on $M,\alpha,c_{H}$ and $T.$ The sequence $(u_{n})_{n\in\mathbb{N}}$ thus converges in $MR_{2}(V,V^{\prime})\cap C([0,T],H)$ to $u$ uniformly on the data $f,u_{0}.$

Proof.

For simplicity, we will in the sequel denote all positive constants depending on $M,\alpha,c_{H}$ and $T$ by $\mathbf{c}>0.$ In view of the above Remark, it suffices to prove the first inequality. To that purpose, consider the unbounded linear operators $\mathcal{A},\mathcal{A}_{n}$ and $\mathcal{B}$ with domains $D(\mathcal{A})=D(\mathcal{A}_{n})=L^{2}(0,T;V)$ and D(\mathcal{B})=\big{\{}u\in H^{1}(0,T;V^{\prime}),u(0)=0\big{\}} defined by

[TABLE]

Thus the Cauchy problem (2), respectively (5), has $L^{2}-$maximal regularity in $V^{\prime}$ if and only if the unbounded operator $\mathcal{A}+\mathcal{B},$ respectively $\mathcal{A}_{n}+\mathcal{B},$ with domain

[TABLE]

is invertible. Consider first the case where $u_{0}=0.$ Then we have $u=(\mathcal{A}+\mathcal{B})^{-1}f$ and $u_{n}=(\mathcal{A}_{n}+\mathcal{B})^{-1}f.$ From Theorem 0.1 and $\bf(H_{0})$ we have

[TABLE]

Let now $0\neq u_{0}\in H.$ Choose $\vartheta\in MR_{2}(V,V^{\prime})$ such that $\vartheta(0)=u_{0}$ and $\|\vartheta\|_{MR_{2}(V,V^{\prime})}\leq 2\|u_{0}\|_{H}.$ Set $g_{n}:=-\dot{\vartheta}(\cdot)-\mathcal{A}_{n}(\cdot)\vartheta(\cdot)+f(\cdot)$ and $g:=-\dot{\vartheta}(\cdot)-\mathcal{A}(\cdot)\vartheta(\cdot)+f(\cdot)\in L^{2}(0,T;V^{\prime}).$ Then there exist $v_{n},u\in MR_{2}(V,V^{\prime})$ such that

[TABLE]

and

[TABLE]

By the uniqueness of solvability, $u_{n}=v_{n}+\vartheta$ and $u=v+\vartheta.$ Therefore, using the result of the first part of the proof we obtain

[TABLE]

This completes the proof. ∎

Recall that the non-autonomous Cauchy problem (2) is said to have $L^{2}$-maximal regularity in $H$ if for each $u_{0}\in V$ and $f\in L^{2}(0,T;H)$ the solution $u$ belongs to $MR_{2}(V,H):=L^{2}(0,T;V)\cap H^{1}(0,T;H).$ The next results is an easy consequence of Theorem 1.1.

Corollary 1.2**.**

Assume that the approximation problems 5 has $L^{2}$-maximal regularity in $H.$ Let $u\in V$ and $f\in L^{2}(0,T;H)$ and let $(u_{n})_{n\in\mathbb{N}}$ the sequence of solutions of (5). If $(u_{n})_{n\in\mathbb{N}}$ converges weakly in $MR_{2}(V,H),$ then the limit problem (2) has also $L^{2}$-maximal regularity in $H$ and $u$ is equal to the weak limits of $(u_{n})_{n\in\mathbb{N}}.$

2. $L^{2}$-maximal regularity in $H:$ a weak approximation

Let $\mathfrak{a},\mathfrak{a}_{n}:[0,T]\times V\times V\longrightarrow\mathbb{C}$ be a closed non-autonomous forms satisfying (1) with the same constants $\beta,\alpha$ and $M>0.$ In this section we assume that there exist $0\leq\gamma<1,$ a sequence of non-decreasing continuous function $\omega_{n}:[0,T]\longrightarrow[0,+\infty),n\in\mathbb{N},$ and zero real sequence $(d_{n})_{n\in\mathbb{N}}$ such that the following assumptions hold.

[TABLE]

where $V_{\gamma}:=[H,V]_{\gamma}$ is the complex interpolation space. Note that

[TABLE]

with continuous embeddings. Remark that condition $\bf(H_{2})$ implies that $\mathcal{A}_{n}(t)-\mathcal{A}_{n}(s)\in\mathcal{L}(V,V_{\gamma}^{\prime})$ and

[TABLE]

The following proposition is of great interest for this paper.

Proposition 2.1**.**

[5, Section 2]** Let $b$ be any sesquilinear form that satisfies (1) with the same constants $M$ and $\alpha$ and let $\gamma\in[0,1[.$ Let $\mathcal{B}$ and $B$ be the associated operators on $V^{\prime}$ and $H,$ respectively. Then there exists a constant $c>0$ which depends only on $M,\alpha,\gamma$ and $c_{H}$ such that

1. (1)

$\displaystyle\|(\lambda-\mathcal{B})^{-1}\|_{\mathcal{L}(V_{\gamma}^{\prime},H)}\leq\frac{c}{(1+\,|\,\lambda\,|\,)^{1-\frac{\gamma}{2}}},$** 2. (2)

$\displaystyle\|(\lambda-\mathcal{B})^{-1}\|_{\mathcal{L}(V)}\leq\frac{c}{1+\,|\,\lambda\,|\,},$** 3. (3)

$\displaystyle\|(\lambda-\mathcal{B})^{-1}\|_{\mathcal{L}(H,V)}\leq\frac{c}{(1+\,|\,\lambda\,|\,)^{\frac{1}{2}}},$** 4. (4)

$\displaystyle\|(\lambda-\mathcal{B})^{-1}\|_{\mathcal{L}(V^{\prime},H)}\leq\frac{c}{(1+\,|\,\lambda\,|\,)^{\frac{1}{2}}},$** 5. (5)

$\displaystyle\|(\lambda-\mathcal{B})^{-1}\|_{\mathcal{L}(V^{\prime}_{\gamma},V)}\leq\frac{c}{(1+\,|\,\lambda\,|\,)^{\frac{1-\gamma}{2}}},$** 6. (6)

$\displaystyle\|e^{-s\mathcal{B}}\|_{\mathcal{L}(V_{\gamma}^{\prime},H)}\leq\frac{c}{s^{\gamma/2}},$** 7. (7)

$\displaystyle\|e^{-sB}\|_{\mathcal{L}(V_{\gamma}^{\prime},V)}\leq\frac{c}{s^{\frac{1+\gamma}{2}}},$** 8. (8)

$\displaystyle\|e^{-s\mathcal{B}}\|_{\mathcal{L}(V^{\prime},V)}\leq\frac{c}{s^{\frac{1}{2}}},$** 9. (9)

$\displaystyle\|Be^{-sB}\|_{\mathcal{L}(H)}\leq\frac{c}{s},$** 10. (10)

$\displaystyle\|e^{-sB}\|_{\mathcal{L}(V)}\leq c$**

for each $t\in[0,T],s\geq 0$ and $\lambda\notin\Sigma_{\theta}:=\{re^{i\varphi}:r>0,|\varphi|<\theta\}.$

Remark 2.2**.**

All estimates in Proposition 2.1 holds for $\mathcal{A}_{n}(t)$ and $\mathcal{A}(t)$ with constant independent of $n$ and $t\in[0,T],$ since $\mathfrak{a}_{n}\mathfrak{a}$ satisfies (1) with the same constants $M,\beta$ and $\alpha,$ also $\gamma$ and $c_{H}$ does not depend on $n$ and $t\in[0,T].$

Notation 2.3**.**

To keep notations simple as possible we will in the sequel denote all positive constants depending on $M,\alpha,\gamma,c_{H}$ and $T$ that appear in proofs and theorems uniformly as $\mathbf{c}>0.$

For each $f\in L^{2}(0,T;H)$ and $u_{0}\in V,$ the solutions $u_{n},n\in\mathbb{N},$ of (5) satisfies the following key formula

[TABLE]

for all $t\in[0,T].$ This formula is due to Acquistapace and Terreni [1] and was proved in a more general setting in [5, Proposition 3.5]. In the sequel we will use the following notations:

[TABLE]

With this notation we can state the main result of this section which, in particular, shows that the limit problem (2) also has $L^{2}$-maximal regularity in $H.$

Theorem 2.4**.**

Assume that the assumptions $\bf(H_{1})$-$\bf(H_{4})$ holds. Then the problem (2) also has $L^{2}$-maximal regularity in $H.$ Moreover, if $f\in L^{2}(0,T;H)$ and $u_{0}\in V$ and $(u_{n})_{n\in\mathbb{N}}\subset MR_{2}(V,H)$ is the sequence of the unique solutions of (5), then $(u_{n})_{n\in\mathbb{N}}$ converges weakly in $MR_{2}(V,H)$ and $u:={\rm w}-\lim\limits_{n\to\infty}u_{n}$ satisfies (2).

For the proof we need first some preliminary lemmas. Using the same argument as in the proof of [5, Theorem 4.1], the next two lemmas follow thanks to $\bf(H_{1})$-$\bf(H_{3})$ and Remark 2.2.

Lemma 2.5**.**

Assume that the assumptions $\bf(H_{1})$-$\bf(H_{4})$ holds. Let $Q_{n}^{\mu}:L^{2}(0,T;H)\to L^{2}(0,T;H)$ denotes the linear operator defined for all $g\in L^{2}(0,T;H)$ and $\mu\geq 0$ by

[TABLE]

Then $\lim\limits_{\mu\to\infty}\|Q_{n}^{\mu}\|_{L^{2}(0,T;H)}=0$ uniformly on $n$ and thus $I-Q_{n}^{\mu}$ is invertible on $L^{2}(0,T;H)$ for $\mu$ large enough and for all $n.$

Lemma 2.6**.**

Assume that the assumptions $\bf(H_{1})$-$\bf(H_{3})$ holds. The following tow estimates

[TABLE]

hold.

Now we can give the proof of Theorem 2.4.

Proof.

(of Theorem 2.4) According to Lemma 2.5 and replacing $\mathcal{A}_{n}(t)$ with $\mathcal{A}_{m}(t)_{n}+\mu,$ we may assume without loss of generality that $Q_{n}=Q^{\mu}_{n}$ satisfies $\|Q_{n}\|_{\mathcal{L}(L^{2}(0,T;H))}<1,$ and then $I-Q_{n}$ is invertible by the Neumann series. We deduce from (7) that

[TABLE]

This equality and Lemma 2.6, yield the estimate

[TABLE]

Since for all $t\in[0,T]$ one has $u_{n}(t)=u_{n}(0)+\int_{0}^{t}\dot{u}_{n}(s){\rm d}s,$ we conclude that

[TABLE]

Then there exists a subsequence of $(u_{n}),$ still denoted by $(u_{n})$ that converges weakly to some $v\in H^{1}(0,T;H)$

On the other hand, the Cauchy problem (2) has a unique solution $u\in MR_{2}(V,V^{\prime}),$ and $(u_{n})$ converges strongly to $u$ on $MR_{2}(V,V^{\prime})$ by Theorem 1.1. We conclude by uniqueness of limits that $u=v\in H^{1}(0,T;H).$ This completes the proof. ∎

3. $L^{2}$-maximal regularity in $H:$ uniform approximation

Assume that $\mathfrak{a}$ and $\mathfrak{a}_{n}$ are as in Section 2. Let $(f,u_{0})\in L^{2}(0,T;H)\times V$ and let $u,u_{n}\in MR_{2}(V,H)$ be the solutions of (2) and (5), respectively. In the previous section we have seen that $(u_{n})_{n\in\mathbb{N}}$ converges weakly to $u$ with respect to the norm of $MR_{2}(V,H).$ The aim of this section is to prove that this convergence holds for the strong topology of $MR_{2}(V,H)$ and uniformly on the initial data $u_{0}$ and $f.$ To this end, we impose the following additional conditions:

[TABLE]

Recall that $-A_{n}(t)$ generates a holomorphic $C_{0}$-semigroup (of angle $\theta:=\frac{\pi}{2}-\arctan(\frac{M}{\alpha})$) $e^{-sA_{n}(t)}$ on $H$ which is the restriction to $H$ of $e^{-\cdot A_{n}(t)},$ and we have

[TABLE]

where $\Gamma:=\{re^{\pm\varphi}:\ r>0\}$ for some fixed $\varphi\in(\theta,\frac{\pi}{2}).$

Theorem 3.1**.**

Assume that the assumptions $\bf(H_{1})$-$\bf(H_{6})$ holds. Then there exists a positive constant $\textbf{c}>0$ depending only on $M,\alpha,\gamma$ and $c_{H}$ such that

[TABLE]

Thus $(u_{n})_{n\in\mathbb{N}}$ convergences to $u$ for the strong topology of $MR_{2}(V,H)$ and uniformly on the initial data $u_{0}$ and $f.$

Proof.

We only have to prove (13) the uniform convergence with respect to $u_{0},f$ in $MR_{2}(V,H)$ becomes obvious. Indeed, we known from Theorem 1.1 that $u_{n}\longrightarrow u$ in $L^{2}(0,T;V)$ uniformly on the initial data $u_{0}$ and the homogeneity $f.$

We will use the representation formula (7) and (8). We proceed by several steps. Let $m,k\in\mathbb{N}$ and set $n:=m+k$ and $d_{n,m}:=d_{n}+d_{m}.$

$(a)$ First, we estimate $\mathcal{A}_{n}u_{n,1}-\mathcal{A}_{m}u_{m,1}$ in $L^{2}(0,T;H).$ Let $t\neq 0.$ Using $\bf(H_{1})$ we obtain and the estimates (6) and (10) in Proposition 2.1 that

[TABLE]

Similarly, combining the estimates (1) and (3) in Proposition 2.1 and the estimate (35) in Proposition 2.1 we obtain

[TABLE]

The last integral is well defined since the function $h:\mathbb{R}\to\mathbb{R}$ given by $h(t)=t^{-\frac{1+\gamma}{2}}$ for $t\in]0,T]$ and $h(t)=0$ for $t\in]-\infty,0]\cap]T,+\infty[$ belongs to $L^{1}(\mathbb{R})$ because $\frac{1+\gamma}{2}<1.$ The estimates (14) and (15) yield, respectively,

[TABLE]

and

[TABLE]

$(b)$ Next, we prove the following estimate

[TABLE]

where $Q_{m}:L^{2}(0,T;H)\longrightarrow L^{2}(0,T;H)$ is defined (9). To this end, for $g\in L^{2}(0,T;H)$ and $t\in[0,T]$ we write

[TABLE]

Replacing $\mathcal{A}_{m}(s)$ by $\mathcal{A}_{m}(s)+\mu$ and according to Proposition 2.1 we may assume $\|\mathcal{A}^{-1}_{n}(s)\|_{\mathcal{L}(V_{\gamma}^{\prime},V)}\leq\mathbf{c}$ and $\|\mathcal{A}^{-1}_{n}(s)\|_{\mathcal{L}(H,V)}\leq\mathbf{c}.$ Next, by the estimates (6) and (9) in Proposition 2.1 together with $\bf(H_{1})-\bf(H_{2})$, we obtain

[TABLE]

where $h_{n}(t):=\omega_{n}(t)t^{-1-\gamma/2}$ for $t\in[0,T]$ and $h_{n}(t):=0$ for $t\in(-\infty,0[\cap]T,+\infty).$ The assumption $\bf(H_{3})$ implies that $h_{n}\in L^{1}(\mathbb{R})$ and that $(\|h_{n}\|_{L^{1}(\mathbb{R})})_{n\in\mathbb{N}}$ is bounded. Therefore, we obtain

[TABLE]

Again by estimates (6) and (9) in Proposition 2.1, we obtain for the second term $I_{n,m,2}$

[TABLE]

where

[TABLE]

Thanks to $\bf(H_{3})$ and $\bf(H_{5})$, $t\mapsto\kappa_{n,m}(t)$ belongs to $L^{1}(\mathbb{R})$, and by a simple calculation we obtain

[TABLE]

Therefore,

[TABLE]

For the last term $I_{n,m,3}(t),$ we set $\tilde{g}_{m}(t,\cdot):=(\mathcal{A}_{m}(t)-\mathcal{A}_{m}(\cdot))A_{m}^{-1}(\cdot)g(\cdot).$ Again by assumptions $\bf(H_{1})$ and (4) and (5) from Proposition 2.1 and we obtain

[TABLE]

Next, since

[TABLE]

it follows

[TABLE]

Using the same argument as that used above for (17) one obtain

[TABLE]

This together with 19 and (17) give the desired estimate (16).

$(c)$ Using Lemma 2.6 we conclude from $a)-b)$ that

[TABLE]

Finally, since $(u_{n})_{n\in\mathbb{N}}$ satisfies (5), we conclude that

[TABLE]

with

[TABLE]

Thus $(\dot{u}_{n})$ is a Cauchy sequence. The limits coincides with $u$ according to Corollary (1.1). This completes the proof of (13) by taking $k\longrightarrow\infty$ in (21).

∎

4. Uniform approximation on $C(0,T;V)$

Let $\mathfrak{a},\mathfrak{a}_{n}:[0,T]\times V\times V\longrightarrow\mathbb{C}$ are as in Section 3. Additionally, we assume that $(u_{n})_{n}\subset C([0,T],V).$ Then we show in this section that $(u_{n})_{n}$ converges in $C([0,T],V)$ uniformly on $(f,u_{0}).$

Proposition 4.1**.**

With the notations of Section 3 the following estimate holds

[TABLE]

for all $k,m\in\mathbb{N}.$

In view of Theorem (1.1), the following is then true and follows immediately from Proposition (4.1).

Corollary 4.2**.**

Let $(f,u_{0})\in L^{2}(0,T;H)\times V$ and let $u\in MR_{2}(V,H)$ be the solution of (2). Then $u\in C(0,T;V)$ and

[TABLE]

holds.

Proof.

We will proceed similarly as in the proof of Theorem 3.1. Let $m,k\in\mathbb{N}$ and set $n:=m+k$ and $d_{n,m}:=d_{n}+d_{m}.$

Step a: By using (2) and (5) in Proposition 2.1 for $(\lambda-\mathcal{A}_{n}(t))^{-1}$ and $(\lambda-\mathcal{A}_{m}(t))^{-1},$ respectively, and $\bf(H_{1})$ we obtain for every $t\in[0,T]$ that

[TABLE]

Step b: Again the estimates (4) and (5) in Proposition 2.1 and formula $\bf(H_{1})$ imply that

[TABLE]

Therefore, we obtain by using Fubini’s theorem that for all $\lambda\in\Gamma\setminus\{0\}$

[TABLE]

Step c: For each $h\in C(0,T;V)$ we set

[TABLE]

From [5, Lemma 4.5] we have $(P_{n}h)_{n\in\mathbb{N}}\subset C(0,T;V).$ Thanks to Proposition 5.2 and assumptions ${\bf(H_{2})}$-${\bf(H_{3})}$ one can prove in a similar way as in Step 3 of the proof of [5, Theorem 4.4] (see also Step 3 of the proof of Lemma 2.5) that $\|P_{n}\|_{\mathcal{L}(C(0,T;V)}\leq 1/2$ and thus $I-P_{n}$ is invertible on $\mathcal{L}(C(0,T,V)).$ Therefore, we obtain by using the representation formula (7)

[TABLE]

The term on the right hand side of (24) is treated in Step a)-b). We need only to estimate the difference $P_{n}-P_{m}$ on $\mathcal{L}(C(0,T;V)).$ For each $h\in C(0,T;V)$ and $t\in[0,T]$ we have

[TABLE]

From $\bf(H_{1})$-$\bf(H_{2})$ and the estimate (5) in Proposition 2.1 we have

[TABLE]

Thus using $\bf(H_{3})$, it follows

[TABLE]

Next, writing

[TABLE]

then from (7) and (8) in Proposition 2.1 and the fact that $e^{-\cdot A_{n}(t)}$ is an analytic $C_{0}$-semigroup on $V$ we obtain

[TABLE]

where $\kappa_{n,m}$ is defined in the proof of Theorem 3.1. Therefore, using (18) we conclude

[TABLE]

This complete the proof of the proposition.

∎

5. Example: an affine approximation

The aim of this section is to provide an explicit approximation of $\mathfrak{a}$ that satisfies the required hypothesis $\bf(H_{1})-\bf(H_{6}).$ Recall that $V,H$ denote two separable complex Hilbert spaces and $\mathfrak{a}:[0,T]\times V\times V\to\mathbb{C}$ is a non-autonomous closed form satisfying (1). Assume moreover that there exists $0\leq\gamma<1$ and a non-decreasing continuous function $\omega:[0,T]\longrightarrow[0,+\infty)$ with

[TABLE]

[TABLE]

and

[TABLE]

Remark 5.1**.**

We note that, the main example of a continuity modulus $\omega$ introduced above is the function $\omega(t)=t^{\eta}$ where $\eta>\gamma/2.$ This main example moreover satisfies

[TABLE]

For general case, thanks to (27), one can always find a null sequence $(t_{n})_{n\in\mathbb{N}}\subset\mathbb{R}_{+}$ with $\lim\limits_{n\to\infty}{\omega(t_{n})}{t_{n}^{-\gamma/2}}=0.$ This is true because $\displaystyle\liminf_{t\to 0}\omega(t)t^{-\gamma/2}=0,$ since otherwise we would have $\displaystyle\int_{0}^{T}\frac{\omega(s)}{s^{1+\gamma/2}}{\rm d}s=\infty$ which contradict (27).

Now let $\Lambda=(0=\lambda_{0}<\lambda_{1}<...<\lambda_{n+1}=T)$ be a uniform subdivision of $[0,T],$ i.e.,

[TABLE]

and consider a family of sesquilinear forms $\mathfrak{a}_{k}:V\times V\to\mathbb{C}$ given by

[TABLE]

for each $k=0,1,...,n.$ Remark that $\mathfrak{a}_{k}$ satisfies (1) for all $k=0,1,...n.$ Then $\mathfrak{a}_{\Lambda}:[0,T]\times V\times V\to\mathbb{C}$ defined for $t\in[\lambda_{k},\lambda_{k+1}]$ by

[TABLE]

is a non-autonomous closed sesquilinear forms satisfying (1) with the same constants $\alpha,\beta$ and $M.$ The associated time dependent operator is denoted by

[TABLE]

and is given for $t\in[\lambda_{k},\lambda_{k+1}]$ by

[TABLE]

where

[TABLE]

In what follows we extend $\omega$ to $[0,2T]$ by setting $\omega(t)=\omega(T)$ for $T\leq t\leq 2T.$

Proposition 5.2**.**

For all $u,v\in V,\ t,s\in[0,T]$ we have

[TABLE]

where $\omega_{\Lambda}:[0,T]\longrightarrow[0,+\infty[$ is defined by

[TABLE]

Moreover, $\mathcal{A}_{\Lambda}(t)-\mathcal{A}_{\Lambda}(s)\in\mathcal{L}(V,V_{\gamma}^{\prime}),$

[TABLE]

and

[TABLE]

for each $t,s\in[0,T].$

Proof.

Let $u,v\in V$ and $t,s\in[0,T].$ For the proof of (33) we distinguish three cases

Case 1: If $\lambda_{k}\leq s<t\leq\lambda_{k+1}$ for some fixed $k\in\{0,1,\cdots,n\}.$ Then we obtain, using (28) and the fact that $\omega$ is non-decreasing, that

[TABLE]

Case 2: If $\lambda_{k}\leq s\leq\lambda_{k+1}\leq t\leq\lambda_{k+2},$ then we deduce from Step 1 that

[TABLE]

Case 3: If now $\lambda_{k}\leq s\leq\lambda_{k+1}<\cdots<\lambda_{l}\leq t\leq\lambda_{l+1}.$ Then $\lambda_{l}-\lambda_{k+1}\leq t-s\leq\lambda_{l+1}-\lambda_{k}$ and thus

[TABLE]

It follows that

[TABLE]

Because of (36) and since $\lambda_{k}-\lambda_{l}=\lambda_{k+1}-\lambda_{l+1},$ we deduce that

[TABLE]

This completes the proof of (33). Next, (34) follows from (33). For the second statement, let $t\in[0,T]$ and let $k\in\{0,1,\cdots,n\}$ be such that $t\in[\lambda_{k},\lambda_{k+1}].$ Then

[TABLE]

Then using (28) and the fact that $\omega$ is non-decreasing we obtain

[TABLE]

which proves the claim. ∎

Recall that a coercive and bounded form $b:V\times V\to\mathbb{C}$ associated with the operator $B$ on $H$ has the Kato square root property if

[TABLE]

We prove in Proposition 5.3 below that $\mathfrak{a}_{\Lambda}(t,\cdot,\cdot)$ has the square root property for all $t\in[0,T]$ if $\mathfrak{a}_{\Lambda}(0;\cdot,\cdot)$ has it. This is essentially based on the abstract result due to Arendt and Monniaux [5, Proposition 2.5]. They proved that for two sesquilinear forms $\mathfrak{a}_{1},\mathfrak{a}_{2}:V\times V\to\mathbb{C}$ which satisfies (1), the form $\mathfrak{a}_{1}$ has the square root property if and only if $\mathfrak{a}_{2}$ has it provided that

[TABLE]

for some constant $c>0.$

Proposition 5.3**.**

Assume $\mathfrak{a}(0,.,.)$ has the square root property. Then $\mathfrak{a}_{\Lambda}(t,.,.)$ has the square root properties for all $t\in[0,T],$ too.

Proof.

Let $t\in[0,T]$ and let $k\in\{0,1,\cdots,n\}$ be such that $t\in[\lambda_{k},\lambda_{k+1}].$ Then Then assumption (28) implies that

[TABLE]

Now the claim follows from [5, Proposition 2.5].∎

Let $\mathcal{A}_{\Lambda}$ be given by (32) and consider the Cauchy problem

[TABLE]

Next, we use the above results to prove that $\mathfrak{a}_{\Lambda}$ satisfies all assumption $\bf(H_{1})$-$\bf(H_{6})$ by taking $d_{n}=\omega(2|\Lambda|)=\omega(2T/n)$ and $\omega_{n}(\cdot)=\omega_{\Lambda}(\cdot).$

Proposition 5.4**.**

Assume that $\mathfrak{a}$ satisfies (26)-(28) and that $\mathfrak{a}(0,\cdot\cdot)$ has the square properties. Then $\mathfrak{a}_{\Lambda}$ satisfies assumptions $\bf(H_{1})$-$\bf(H_{5}),$ and satisfies also $\bf(H_{6})$ if moreover (29) holds. Furthermore, the solution $u_{\Lambda}$ of (38) belongs to $C(0,T;V)$ for each given $u_{0}\in V$ and $f\in L^{2}(0,T;H).$

Proof.

According to Proposition 5.2, $\mathfrak{a}_{\Lambda}$ satisfies $\bf(H_{1})$ and $\bf(H_{2}).$ By the definition of $\omega_{\Lambda}$ it follows

[TABLE]

which is finite by (26) and (27). Next, it is easy to prove that

[TABLE]

holds. On the other hand, the function $t\mapsto\mathfrak{a}_{\Lambda}(\cdot,u,v)$ is piecewise $C^{1}$ for all $u,v\in V$ and $\mathfrak{a}_{\Lambda}(t,\cdot,\cdot),t\in[0,T],$ has the Kato square property by Lemma 5.3. Then the Cauchy problem (38) has $L^{2}-$maximal regularity in $H$ and $u_{\Lambda}\in C(0,T;V)$ for each $u_{0}\in V$ and $f\in L^{2}(0,T;H)$ [4, Theorem 4.2]. Therefore, $\bf(H_{3})$ and $\bf(H_{4})$ are also satisfied by $\mathfrak{a}.$ Assume now that $(\ref{model contiuity additional})$ holds. Then we obtain that $\displaystyle{\omega(2|\Lambda|)}{|\Lambda|^{-\gamma/2}}$ and

[TABLE]

converge to [math] as $|\Lambda|\longrightarrow 0.$ Thus the fact that $\omega$ is non-decreasing complete the proof.

∎

The next provides in particular an alternative proof of some results in [5].

Corollary 5.5**.**

Assume that $\mathfrak{a}$ satisfies (26)-(28) and that $\mathfrak{a}(0,\cdot\cdot)$ has the square properties. Then (38) has $L^{2}$-maximal regularity in $H$ and for each $u_{0}\in V$ and $f\in L^{2}(0,T;H)$ the solution $(u_{\Lambda})_{\Lambda}$ converges weakly in $MR_{2}(V,H)$ as $|\Lambda|\longrightarrow 0,$ and $u:=w-\lim\limits_{|\Lambda|\to 0}u_{\Lambda}$ satisfies (2). If moreover, $(\ref{model contiuity additional})$ holds then $u_{\Lambda}\longrightarrow u$ strongly in $MR_{2}(V,H)\cap C(0,T;V)$ and uniformly on $(u_{0},f)$ as $|\Lambda|\longrightarrow 0.$ Further, the following estimates

[TABLE]

and

[TABLE]

holds.

Proof.

The proof follows from Theorem (3.1), Proposition 5.4 and Corollary 4.2 ∎