Well-posedness of a Pseudo-Parabolic KWC System
in Materials Science ¹¹footnotemark: 1

The Kobayashi–Warren–Carter (KWC) system consists of a set of non-smooth parabolic PDEs and is widely used in materials science [14, 15]. It is based on the phase field model of planar grain boundary motion. This model suffers from two fundamental challenges: (i) Physics: It is difficult to establish KWC-system as the gradient flow of a variational model; (ii) Mathematics: The solution to KWC-system are known to be unique only under special cases. Both of these challenges makes it difficult to rigorously use this model in practice or carry our new material design via optimization [5, 6].

This article aims to overcome both these challenges by introducing a pseudo-parabolic version of the KWC-system. Well-posedness (both existence and uniqueness) of the resulting system (S), which arise from gradient flow based on the KWC-energy

is established. Namely, this article addresses open issues from previous works that deal with the KWC-system (cf. [18, 19, 22, 20, 21, 17]) and its regularized versions (cf. [13, 24, 7, 5, 6, 16]).

Next, we describe the system (S). Let $0<T<\infty$ be a fixed final time, and let $N\in\{1,2,3\}$ denote the spatial dimension. Let $\Omega\subset\mathbb{R}^{N}$ be an open bounded spatial domain with a boundary $\Gamma:=\partial\Omega$. When $N>1$, $\Gamma$ is assumed to be sufficiently smooth, with the unit outer normal $n_{\Gamma}$. Besides, we let $Q:=(0,T)\times\Omega$ and $\Sigma:=(0,T)\times\Gamma$. Then the pseudo-parabolic system denoted by (S), with two constants $\mu\geq 0$ and $\nu>0$, is given by

Here, the unknowns $\eta=\eta(t,x)$ and $\theta=\theta(t,x)$ are order parameters that indicate the orientation order and orientation angle of the polycrystal body, respectively. Besides, $\eta_{0}=\eta_{0}(x)$ and $\theta_{0}=\theta_{0}(x)$ is the initial data. Moreover, $u=u(t,x)$ and $v=v(t,x)$ are the forcing terms. Additionally, $\alpha_{0}=\alpha_{0}(\eta)$ and $\alpha=\alpha(\eta)$ are fixed positive-valued functions to reproduce the mobilities of grain boundary motions. Finally, $g=g(\eta)$ is a perturbation for the orientation order $\eta$, having a nonnegative potential $G=G(\eta)$, i.e. $\frac{d}{d\eta}G(\eta)=g(\eta)$.

A generic form of the “KWC-system” is given by the evolution equation (cf. [14, 15]):

which is motivated by the gradient flow of the free-energy, namely the KWC-energy (1.1), with a functional derivative $\frac{\delta}{\delta[\eta,\theta]}\mathcal{F}$, and an unknown-dependent monotone operator $\mathcal{A}_{0}(\eta)\subset[L^{2}(\Omega)]^{2}$. Here, the evolution equation (1.2) can be considered as the common root of the original KWC-system (cf. [14, 15]) and our system (S). Indeed, our system (S) is derived from the evolution equation (1.2), in the case when:

while the original KWC-system corresponds to the case when $\mu=\nu=0$.

In recent years, the principal issue has been to clarify the variational structure (representation) of the functional derivative $\frac{\delta}{\delta[\eta,\theta]}\mathcal{F}$ of the nonsmooth and nonconvex energy $\mathcal{F}$ in (1.1). The positive answer to the issue was obtained in [18, 19, 22], by means of BV-theory (cf. [4, 11, 12, 8, 2, 3]), and this work has provided a basis of the study of KWC-system, e.g. the existence and large-time behavior [18, 19], the observations under other boundary conditions [20, 21], the time-periodic solution [17], and so on.

However, the uniqueness of solutions has been a significant challenge, due to the velocity term $\alpha_{0}(\eta)\partial_{t}\theta$ and the singular diffusion flux $\alpha(\eta)\frac{D\theta}{|D\theta|}$, both of which depend on the unknown-dependent mobilities. Therefore, previous researchers have implemented the following modifications to the modelling framework (1.1) and (1.3):

$\bullet$

resetting $\alpha_{0}$ to be a function which is independent of $\eta$ (effectively a constant);

$\bullet$

modifying the free-energy functional to a more relaxed form:

These modifications have been pivotal in addressing the uniqueness challenges (cf. [13, 24]), and several advanced issues, such as the optimal control problems (see [7, 5, 6, 16]).

In light of this, we can expect that the pseudo-parabolic nature of our system will effectively address the uniqueness challenge. This is due to the positive constants $\mu$ and $\nu$ in (1.3), which are expected to bring a smoothing effect for the regularity of solution. In addition, it is also crucial from a mathematical perspective to clarify the similarities and differences between our pseudo-parabolic system and the original parabolic KWC-system.

Consequently, we set the goal to prove the following two Main Theorems, concerned with the well-posedness of our pseudo-parabolic system (S), i.e. the evolution equation (1.2) under (1.1) and (1.3).

Main Theorem 1.

Existence and regularity of solution to (S).

Main Theorem 2.

Uniqueness of solution to (S), and continuous dependence with respect to the initial data and forcings.

These Main Theorems will provide the positive answer to our earlier expectation regarding the effectiveness of the pseudo-parabolic nature of our system in resolving the uniqueness issue. Also, the Main Theorems will focus on two conflicting properties: the singularity in the diffusion flux $\alpha(\eta)\frac{D\theta}{|D\theta|}$; and the smoothing effect encouraged by the Laplacian in (1.3). This conflicting situation will be clarified by the differences in regularity between components: $\eta\in W^{1,2}(0,T;H^{2}(\Omega))$; and $\theta\in W^{1,2}(0,T;H^{1}(\Omega))\cap L^{\infty}(0,T;H^{2}(\Omega))$; in the Main Theorem 1. Moreover, the results of this paper will form a fundamental part of the optimization problem in grain boundary motion, which will be exploring in a forthcoming paper.

We begin by prescribing the notations used throughout this paper.

Let $d\in\mathbb{N}$ be a fixed dimension. We denote by $|y|$ and $y\cdot z$ the Euclidean norm of $y\in\mathbb{R}^{d}$ and the scalar product of $y,z\in\mathbb{R}^{d}$, respectively, i.e.,

Besides, we let:

We denote by $\mathcal{L}^{d}$ the $d$-dimensional Lebesgue measure, and we denote by $\mathcal{H}^{d}$ the $d$-dimensional Hausdorff measure. In particular, the measure theoretical phrases, such as “a.e.”, “$dt$”, and “$dx$”, and so on, are all with respect to the Lebesgue measure in each corresponding dimension. Also on a Lipschitz-surface $S$, the phrase “a.e.” is with respect to the Hausdorff measure in each corresponding Hausdorff dimension. In particular, if $S$ is $C^{1}$-surface, then we simply denote by $dS$ the area-element of the integration on $S$.

For a Borel set $E\subset\mathbb{R}^{d}$, we denote by $\chi_{E}:\mathbb{R}^{d}\longrightarrow\{0,1\}$ the characteristic function of $E$. Additionally, for a distribution $\zeta$ on an open set in $\mathbb{R}^{d}$ and any $i\in\{1,\dots,d\}$, let $\partial_{i}\zeta$ be the distributional differential with respect to $i$-th variable of $\zeta$. As well as we consider, the differential operators, such as $\nabla,\ \operatorname{div},\ \nabla^{2}$, and so on, in distributional sense.

For two Banach spaces $X$ and $Y$, let $\mathscr{L}(X;Y)$ be the Banach space of bounded linear operators from $X$ into $Y$.

For Banach spaces $X_{1},\dots,X_{d}$ with $1<d\in\mathbb{N}$, let $X_{1}\times\dots\times X_{d}$ be the product Banach space endowed with the norm $|\cdot|_{X_{1}\times\cdots\times X_{d}}:=|\cdot|_{X_{1}}+\cdots+|\cdot|_{X_{d}}$. However, when all $X_{1},\dots,X_{d}$ are Hilbert spaces, $X_{1}\times\dots\times X_{d}$ denotes the product Hilbert space endowed with the inner product $(\cdot,\cdot)_{X_{1}\times\cdots\times X_{d}}:=(\cdot,\cdot)_{X_{1}}+\cdots+(\cdot,\cdot)_{X_{d}}$ and the norm $|\cdot|_{X_{1}\times\cdots\times X_{d}}:=\bigl{(}|\cdot|_{X_{1}}^{2}+\cdots+|\cdot|_{X_{d}}^{2}\bigr{)}^{\frac{1}{2}}$. In particular, when all $X_{1},\dots,X_{d}$ coincide with a Banach space $Y$, the product space $X_{1}\times\dots\times X_{d}$ is simply denoted by $[Y]^{d}$.

In the meantime, for any $q\in[1,\infty)$ and any $\zeta\in L_{\mathrm{loc}}^{q}([0,\infty);X)$, we denote by $\{\zeta_{i}\}_{i=0}^{\infty}\subset X$ the sequence of time-discretization data of $\zeta$, defined as:

In this paper, the main assertions are discussed under the following assumptions.

• (A1)

  $\mu>0$ and $\nu>0$ are fixed constants.

• (A2)

  $g:\mathbb{R}\longrightarrow\mathbb{R}$ is a locally Lipschitz continuous function with a nonnegative primitive $G\in C^{1}(\mathbb{R})$. Moreover, $g$ satisfies the following condition:

  $$\liminf_{\xi\downarrow-\infty}g(\xi)=-\infty,\ \limsup_{\xi\uparrow\infty}g(\xi)=\infty.$$

• (A3)

  $\alpha_{0}:\mathbb{R}\longrightarrow(0,\infty)$ is a locally Lipschitz continuous function, and $\alpha:\mathbb{R}\longrightarrow[0,\infty)$ is a $C^{1}$-class convex function, such that:

  $$\alpha^{\prime}(0)=0\mbox{ and }\delta_{\alpha}:=\inf\alpha_{0}(\mathbb{R})>0.$$

• (A4)

  $u,v\in L_{\rm loc}^{2}([0,\infty);H)$, and $u\in L^{\infty}(Q)$.

• (A5)

  The initial data $[\eta_{0},\theta_{0}]$ belong to the class $[W_{0}]^{2}\subset[H^{2}(\Omega)]^{2}$.

Now, the main results are stated as follows.

In the Main Theorems, the solution to (S) will be obtained by means of the time-discretization method. In this light, let $\tau\in(0,1)$ be a constant of the time-step size, and let $\varepsilon\in(0,1)$ be a relaxation constant. Based on this, we adopt the following time-discretization scheme (AP)${}_{\tau}^{\varepsilon}$, as our approximating problem of (S).

The solution to (AP)${}_{\tau}^{\varepsilon}$ is given as follows.

In this paper, the following theorem will plays an important role for the proof of Main Theorems.

Theorem 1 is proved through several lemmas.