粘弹性相分离模型在二维空间中强解的整体存在性 The Overall Existence of Strong Solution for Viscoelastic Phase Separation Model in the Two Dimensional Space

doi:10.12677/AAM.2023.126275

Advances in Applied Mathematics
Vol. 12 No. 06 ( 2023 ), Article ID: 67123 , 9 pages
10.12677/AAM.2023.126275

粘弹性相分离模型在二维空间中强解的整体存在性

裴小田

●How to Cite this Article

辽宁师范大学数学学院，辽宁大连

收稿日期：2023年5月13日；录用日期：2023年6月7日；发布日期：2023年6月15日

摘要

本文给出了粘弹性相分离模型在二维空间下强解的整体存在性，使用Gagliardo-Nirenberg不等式、Sobolev不等式和Gronwall不等式进行证明，还使用了先验估计的证明方法。

关键词

粘弹性相分离模型，强解，二维空间

The Overall Existence of Strong Solution for Viscoelastic Phase Separation Model in the Two Dimensional Space

Xiaotian Pei

School of Mathematics, Liaoning Normal University, Dalian Liaoning

Received: May 13^th, 2023; accepted: Jun. 7^th, 2023; published: Jun. 15^th, 2023

ABSTRACT

We present the overall existence of strong solutions for the viscoelastic phase separation model in two dimensions, using Gagliardo-Nirenberg inequality, Sobolev inequality and Gronwall inequality, and also the proof method of prior estimation.

Keywords:Viscoelastic Phase Separation Model, Strong Solution, Two Dimensional Space

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 介绍

本文讨论在二维空间下粘弹性相分离模型强解的存在性

$\frac{\partial ϕ}{\partial t} + u \cdot \nabla ϕ = d i v (m (ϕ) \nabla μ) - κ d i v (n (ϕ) \nabla (A (ϕ) q)),$ (1.1)

$\frac{\partial q}{\partial t} + u \cdot \nabla q = - \frac{1}{τ (ϕ)} q + A (ϕ) Δ (A (ϕ) q) - κ A (ϕ) d i v (n (ϕ) \nabla μ) + ε Δ q,$ (1.2)

$\frac{\partial u}{\partial t} + (u \cdot \nabla) u = d i v (η (ϕ) (\nabla u + {(\nabla u)}^{T})) - \nabla P + \nabla ϕ μ,$ (1.3)

$d i v (u) = 0,$ (1.4)

$μ = - Δ ϕ + f (ϕ),$ (1.5)

其中 $ϕ$ 为聚合物分子的体积分数，q为聚合物相互作用产生的体应力，u为由溶剂与聚合物速度组成的体积平均速度， $m (ϕ)$ 和 $n (ϕ)$ 代表移动性函数， $τ (ϕ)$ 代表广义弛豫时间， $A (ϕ)$ 代表体模量， $η (ϕ)$ 为粘度， $ε$ 和 $κ$ 是正常数， $f (ϕ) = F^{'} (ϕ)$ 而 $F (ϕ) = a ϕ^{2} {(ϕ - 1)}^{2}$ 且满足 $| F (s) | \leq c_{1} {| s |}^{p} + c_{2}$ ， $| F^{'} (ϕ) | \leq c_{3} {| ϕ |}^{p - 1} + c_{4}$ ， $| F^{″} (ϕ) | \leq c_{5} {| s |}^{p - 2} + c_{6}$ ，对于 $p \geq 2$ ，其中， $F \in C^{2} (R)$ 并且 $c_{i} > 0$ ，对于 $i \in {1, \dots, 6}$ 。

这里 $Ω \in R^{2}$ 是具有利普希茨边界条件的有界域，该模型具有以下初始条件和边界条件

${(ϕ, q, u) |}_{t = 0} = (ϕ_{0}, q_{0}, u_{0}), {\partial_{n} ϕ |}_{\partial Ω} = {\partial_{n} μ |}_{\partial Ω} = {\partial_{n} q |}_{\partial Ω} = 0, {u |}_{\partial Ω} = 0.$

以下是论文中涉及到的符号假设。在后续内容中，将假设所有共函数都是连续正有界的， $0 < τ_{1} \leq τ (s) \leq τ_{2}$ ， $0 < A_{1} \leq A (s) \leq A_{2}$ ， $0 < η_{1} \leq η (s) \leq η_{2}$ 对于 $s \in R$ ，其中 ${‖ τ^{'} ‖}_{\infty} \leq {τ^{'}}_{2}$ ， ${‖ A^{'} ‖}_{\infty} \leq {A^{'}}_{2}$ ， ${‖ A^{″} ‖}_{\infty} \leq {A^{″}}_{2}$ ， ${‖ η^{'} ‖}_{\infty} \leq {η^{'}}_{2}$ 。还需假设函数对于 $m, n$ 是连续正有界的，表示如下 $0 < m_{1} \leq m (s) \leq m_{2}$ ， $0 < n_{1} \leq n (s) \leq n_{2}$ 对于 $s \in R$ ，其中 ${‖ m^{'} ‖}_{\infty} \leq {m^{'}}_{2}$ ， ${‖ n^{'} ‖}_{\infty} \leq {n^{'}}_{2}$ 。

二元流体的相分离是软物质物理学中的一个基本过程。对于牛顿流体来说，这种现象是很普遍的。粘弹性相分离可以用由相位演化的Cahn-Hilliard方程、低流体演化的Navier-Stokes方程和粘弹性构象张量的时间演化组成的耦合系统来描述。其中 $ϕ$ 的相位变量主要受自由能泛函的低梯度控制，并与所涉及过程的热力学密切相关。对于该模型已经有人得出三维空间整体弱解的存在性，而更加复杂的模型如 [1] 中，也已经得出模型二维空间弱解的存在性以及弱解强解唯一性的结论，本文主要是对 [1] 中的模型简化处理后的方程，讨论研究其在二维空间的整体强解的存在性问题，采用对方程进行更高阶的处理方式，故而得到比 [1] 中的解阶数更高的整体强解，这是本文的重要创新点，这样得到的强解识别性更强，但是进行更高阶处理的同时，对后续估计工作增加的难度，这也是本文的主要难点。此外，本文对原方程进行简单的处理，并且增加了控制项，使得后续的计算证明变得更加简便明确。

本文使用的模型类似于 [1] 、 [2] 和 [3] 中的粘弹性相分离模型，主要研究粘弹性考虑了弹性构象张量的时间演化的模糊彼得林模型，而不是经典Oldroyd-B模型。彼得林模型 [2] 可以看作是经典Oldroyd-B模型的非线性推广。因此，本文整体强解的存在性结果也适用于 [3] 和 [4] 中的粘弹性相分离模型。文献 [5] 中研究了强解的良好性，文中使用的迁移率函数和融合共聚物的模型可以在 [6] 和 [7] 中得到详细解释。更多的关于粘弹性相分离模型的研究，请读者参考 [8] [9] [10] 。

主要结论为以下定理

定理1 假设 $(ϕ_{0}, q_{0}, u_{0}) \in H^{2} (Ω) \times H^{1} (Ω) \times H^{1} (Ω)$ ，这里 $Ω$ 是 $R^{2}$ 上的光滑区域，对于任意有限时间 $T > 0$ ，方程(1.1)~(1.5)在区域 $Ω$ 上存在强解 $(ϕ, q, u)$ 。

下面将介绍一些定理证明所用到的定义和引理。

2. 准备工作

以下定义和引理参考 [1] [2]

定义2.1 假设 $(ϕ_{0}, q_{0}, u_{0}) \in H^{2} (Ω) \times H^{1} (Ω) \times H^{1} (Ω)$ ，这里 $Ω$ 是 $R^{2}$ 上光滑区域，若对 $(ϕ, q, u)$ 有 $ϕ \in L^{\infty} (0, T; H^{2} (Ω))$ ， $q \in L^{\infty} (0, T; H^{1} (Ω))$ ， $u \in L^{\infty} (0, T; H^{1} (Ω))$ 。 $(ϕ, q, u)$ 是方程(1.1)~(1.5)的弱解。

定义2.2 假设 $(ϕ_{0}, q_{0}, u_{0}) \in H^{2} (Ω) \times H^{1} (Ω) \times H^{1} (Ω)$ ，这里 $Ω$ 是 $R^{2}$ 上的光滑区域，对于任意有限时间 $T > 0$ ，若 $(ϕ, q, u)$ 满足 $ϕ \in L^{\infty} (0, T; H^{2} (Ω))$ ， $q \in L^{\infty} (0, T; H^{1} (Ω))$ ， $u \in L^{\infty} (0, T; H^{1} (Ω))$ 。 $(ϕ, q, u)$ 是方程(1.1)~(1.5)的强解。

引理2.1 (Sobolev不等式)设p满足 $1 \leq p < n$ ，对于一切 $u \in C_{0}^{\infty} (R^{n})$ ，

${‖ u ‖}_{L^{q}} \leq C {‖ D u ‖}_{L^{p}},$

其中q满足 $\frac{1}{q} = \frac{1}{p} - \frac{1}{n}$ 。

引理2.2 (Gagliardo-Nirenberg不等式)对于 $0 < q, r \leq \infty$ ， $0 < α < 1$ ， $u \in W^{m, r} (Ω)$ ，这里 $Ω$ 是边界光滑的区域，有

${‖ D^{j} u ‖}_{L^{p}} \leq C {‖ D^{m} u ‖}_{L^{r}}^{α} {‖ u ‖}_{L^{q}}^{1 - α},$

其中 $\frac{j}{m} \leq α < 1$ ， $\frac{1}{p} = \frac{j}{n} + α (\frac{1}{r} - \frac{m}{n}) + (1 - α) \frac{1}{q}$ 。

引理2.2 (Gronwall不等式)设 $η$ 是 $[0, T]$ 到 $[0, \infty)$ 的连续函数， $η (t) \geq 0$ 且满足

$η^{'} (t) \leq ϕ (t) η (t) + ψ (t)$ ，有 $η (t) \leq e^{\int_{0}^{t} ϕ (s) d s} [η (0) + \int_{0}^{t} ψ (s) d s]$ 。

3. 二维空间的强解存在性证明

本节中将给出本文中的定理1的证明。

证明：若要证明定理1，则要假设解 $(ϕ, q, u)$ 是光滑的，对(1.1)作用 $Δ$ ，右乘 $Δ ϕ$ ，对(1.2)作用 $\nabla$ ，右乘 $\nabla q$ ，对(1.3)右乘 $- Δ u$ ，得

$\begin{matrix} \frac{1}{2} \frac{d}{d t} \int_{Ω} {| Δ ϕ |}^{2} d x = - \int_{Ω} Δ (u \cdot \nabla ϕ) Δ ϕ d x + \int_{Ω} Δ (d i v (m (ϕ) \nabla μ)) Δ ϕ - \int_{Ω} Δ (κ d i v (n (ϕ) \nabla (A (ϕ) q))) Δ ϕ d x \\ = - \int_{Ω} m (ϕ) {| Δ^{2} ϕ |}^{2} d x - \int_{Ω} Δ (u \cdot \nabla ϕ) Δ ϕ d x + \int_{Ω} Δ (m^{'} (ϕ) \nabla ϕ \cdot \nabla μ + m (ϕ) Δ F^{'} (ϕ)) Δ ϕ d x \\ - \int_{Ω} κ d i v (n (ϕ) \nabla (A (ϕ) q)) Δ^{2} ϕ d x, \end{matrix}$

$\begin{matrix} \frac{1}{2} \frac{d}{d t} \int_{Ω} {| \nabla q |}^{2} d x = - \int_{Ω} \nabla (u \cdot \nabla q) \nabla q d x - \int_{Ω} \nabla (\frac{1}{τ (ϕ)} q) \nabla q d x + \int_{Ω} \nabla (A (ϕ) Δ (A (ϕ) q)) \nabla q d x \\ - \int_{Ω} \nabla (κ A (ϕ) d i v (n (ϕ) \nabla μ)) \nabla q d x + \int_{Ω} \nabla (ε Δ q) \nabla q d x \\ = - \int_{Ω} ε {| Δ q |}^{2} d x - \int_{Ω} \frac{1}{τ (ϕ)} {| \nabla q |}^{2} d x - \int_{Ω} {| A (ϕ) Δ q |}^{2} d x - \int_{Ω} \nabla (u \cdot \nabla q) \nabla q d x \\ - \int_{Ω} \nabla (\frac{1}{τ (ϕ)} q) \nabla q d x - \int_{Ω} A (ϕ) A^{″} (ϕ) {(\nabla ϕ)}^{2} q Δ q d x - \int_{Ω} A (ϕ) A^{'} (ϕ) Δ ϕ q Δ q d x \\ - \int_{Ω} 2 A (ϕ) A^{'} (ϕ) \nabla ϕ \cdot \nabla q Δ q d x - \int_{Ω} \nabla (κ A (ϕ) d i v (n (ϕ) \nabla μ)) \nabla q d x, \end{matrix}$

$\begin{matrix} \frac{1}{2} \frac{d}{d t} \int_{Ω} {| \nabla u |}^{2} d x = \int_{Ω} (u \cdot \nabla) u (- Δ u) d x + \int_{Ω} d i v (η (ϕ) D u) (- Δ u) d x - \int_{Ω} \nabla P (- Δ u) d x + \int_{Ω} \nabla ϕ μ (- Δ u) d x \\ = - \int_{Ω} η (ϕ) {| Δ u |}^{2} d x + \int_{Ω} (u \cdot \nabla) u (- Δ u) d x - \int_{Ω} η^{'} (ϕ) \nabla ϕ \cdot \nabla u Δ u d x - \int_{Ω} \nabla ϕ μ Δ u d x, \end{matrix}$

其中 $I_{1} = - \int_{Ω} Δ (u \cdot \nabla ϕ) Δ ϕ d x$ 结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$\begin{matrix} I_{1} = 2 \int_{Ω} (\partial_{k} u \cdot \nabla) \partial_{k} ϕ Δ ϕ d x + \int_{Ω} (u \cdot \nabla) Δ ϕ Δ ϕ d x \\ \leq C {‖ \nabla u ‖}_{L^{2}} {‖ Δ ϕ ‖}_{L^{4}}^{2} \\ \leq C {‖ \nabla u ‖}_{L^{2}} {‖ \nabla ϕ ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + C (ε) {‖ \nabla u ‖}_{L^{2}}^{2} . \end{matrix}$

再令 $I_{2} = \int_{Ω} Δ (m^{'} (ϕ) \nabla ϕ \cdot \nabla μ + m (ϕ) Δ F^{'} (ϕ)) Δ ϕ d x$

$\begin{matrix} I_{2} = \int_{Ω} Δ (m^{'} (ϕ) \nabla ϕ \cdot \nabla μ + m (ϕ) Δ F^{'} (ϕ)) Δ ϕ d x \\ = \int_{Ω} m^{'} (ϕ) \nabla ϕ \cdot \nabla μ Δ^{2} ϕ d x + \int_{Ω} m (ϕ) Δ F^{'} (ϕ) Δ^{2} ϕ d x \\ = I_{21} + I_{22}, \end{matrix}$

其中

$\begin{matrix} I_{21} = \int_{Ω} m^{'} (ϕ) \nabla ϕ \cdot \nabla μ Δ^{2} ϕ d x \\ = \int_{Ω} m^{'} (ϕ) \nabla ϕ \nabla (- Δ ϕ + {| ϕ |}^{p - 1} + C) Δ^{2} ϕ d x \\ \leq \int_{Ω} m^{'} (ϕ) \nabla ϕ (\nabla Δ ϕ) Δ^{2} ϕ d x + \int_{Ω} m^{'} (ϕ) \nabla ϕ \nabla ({| ϕ |}^{3}) Δ^{2} ϕ d x + C \int_{Ω} m^{'} (ϕ) \nabla ϕ Δ^{2} ϕ d x, \end{matrix}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{21}$ 的第一部分进行估计得到

$\begin{array}{l} \int_{Ω} m^{'} (ϕ) \nabla ϕ (\nabla Δ ϕ) Δ^{2} ϕ d x \\ \leq c {‖ Δ^{2} ϕ ‖}_{L^{2}} {‖ \nabla Δ ϕ ‖}_{L^{4}} {‖ Δ^{2} ϕ ‖}_{L^{4}} \\ \leq c {‖ Δ^{2} ϕ ‖}_{L^{2}} {‖ \nabla ϕ ‖}_{L^{2}}^{\frac{1}{2}} {‖ Δ ϕ ‖}_{L^{2}}^{\frac{1}{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{\frac{1}{2}} \\ \leq c {‖ Δ^{2} ϕ ‖}_{L^{2}}^{\frac{3}{2}} {‖ Δ ϕ ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{\frac{1}{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{1}) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

同理，对 $I_{21}$ 的第二部分进行估计得到

$\begin{array}{l} \int_{Ω} m^{'} (ϕ) \nabla ϕ \nabla ({| ϕ |}^{3}) Δ^{2} ϕ d x \\ \leq c {‖ \nabla ϕ ‖}_{L^{4}}^{2} {‖ ϕ ‖}_{L^{\infty}}^{2} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq c {‖ \nabla ϕ ‖}_{L^{2}}^{\frac{3}{2}} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{\frac{1}{2}} {‖ ϕ ‖}_{L^{2}}^{\frac{3}{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{\frac{3}{2}} \\ \leq ε_{2} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{2}) {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

再对 $I_{21}$ 的第三部分进行估计得到

$\begin{array}{l} C \int_{Ω} m^{'} (ϕ) \nabla ϕ Δ^{2} ϕ d x \\ \leq c {‖ \nabla ϕ ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq c {‖ Δ ϕ ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{3} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{3}) {‖ Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

将上述不等式相加可以得到

$I_{21} \leq ε {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε) ({‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2}) .$

对于

$\begin{matrix} I_{22} = \int_{Ω} m (ϕ) Δ F^{'} (ϕ) Δ^{2} ϕ d x \\ \leq \int_{Ω} m (ϕ) (| ϕ | {(\nabla ϕ)}^{2} + {| ϕ |}^{p - 1} Δ ϕ + C) Δ^{2} ϕ d x \\ \leq \int_{Ω} m (ϕ) | ϕ | {(\nabla ϕ)}^{2} Δ^{2} ϕ d x + \int_{Ω} m (ϕ) {| ϕ |}^{2} Δ ϕ Δ^{2} ϕ d x + C \int_{Ω} m (ϕ) Δ^{2} ϕ d x, \end{matrix}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{22}$ 的第一部分进行估计得到

$\begin{array}{l} \int_{Ω} m (ϕ) | ϕ | {(\nabla ϕ)}^{2} Δ^{2} ϕ d x \\ \leq C {‖ ϕ ‖}_{L^{\infty}} {‖ \nabla ϕ ‖}_{L^{4}}^{2} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{1}) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

同理，对 $I_{22}$ 的第二部分进行估计得到

$\begin{array}{l} \int_{Ω} m (ϕ) {| ϕ |}^{2} Δ ϕ Δ^{2} ϕ d x \\ \leq c {‖ ϕ ‖}_{L^{\infty}}^{2} {‖ Δ ϕ ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{2} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{2}) {‖ Δ ϕ ‖}_{L^{2}}^{4}, \end{array}$

将上述不等式相加可以得到

$I_{22} \leq ε {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε) ({‖ Δ ϕ ‖}_{L^{2}} {‖ \nabla Δ ϕ ‖}_{L^{2}} + {‖ Δ ϕ ‖}_{L^{2}}^{4}) .$

令 $I_{3} = - \int_{Ω} κ d i v (n (ϕ) \nabla (A (ϕ) q)) Δ^{2} ϕ d x$ 则

$\begin{matrix} I_{3} = \int_{Ω} κ d i v (n (ϕ) \nabla (A (ϕ) q)) Δ^{2} ϕ d x \\ = \int_{Ω} κ n^{'} (ϕ) \nabla ϕ \nabla (A (ϕ) q) Δ^{2} ϕ d x + \int_{Ω} κ n (ϕ) A^{″} (ϕ) {(\nabla ϕ)}^{2} q Δ^{2} ϕ d x + \int_{Ω} κ n (ϕ) A^{'} (ϕ) Δ ϕ q Δ^{2} ϕ d x \\ + \int_{Ω} 2 κ n (ϕ) A^{'} (ϕ) \nabla ϕ \cdot \nabla q Δ^{2} ϕ d x + \int_{Ω} κ n (ϕ) A (ϕ) Δ q Δ^{2} ϕ d x \\ = I_{31} + I_{32} + I_{33} + I_{34} + I_{35}, \end{matrix}$

对于

$\begin{matrix} I_{31} = \int_{Ω} κ n^{'} (ϕ) \nabla ϕ \nabla (A (ϕ) q) Δ^{2} ϕ d x \\ = κ \int_{Ω} n^{'} (ϕ) A (ϕ) \nabla ϕ \cdot \nabla q Δ^{2} ϕ d x + κ \int_{Ω} n^{'} (ϕ) A^{'} (ϕ) {(\nabla ϕ)}^{2} q Δ^{2} ϕ d x, \end{matrix}$

同理，对 $I_{31}$ 的第一部分进行估计得到

$\begin{array}{l} κ \int_{Ω} n^{'} (ϕ) A (ϕ) \nabla ϕ \cdot \nabla q Δ^{2} ϕ d x \\ \leq κ c {‖ \nabla ϕ ‖}_{L^{4}} {‖ \nabla q ‖}_{L^{4}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2}, \end{array}$

同理，对 $I_{31}$ 的第二部分进行估计得到

$κ \int_{Ω} n^{'} (ϕ) A^{'} (ϕ) {(\nabla ϕ)}^{2} q Δ^{2} ϕ d x \leq κ c {‖ q ‖}_{L^{4}} {‖ {(\nabla ϕ)}^{2} ‖}_{L^{4}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (κ, ε_{1}) {‖ Δ ϕ ‖}_{L^{2}}^{2}$

将上述不等式相加可以得到

$I_{31} \leq 2 ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2} + c (κ, ε_{1}) {‖ Δ ϕ ‖}_{L^{2}}^{2},$

由于 $I_{32}$ 和 $I_{31}$ 的第一部分估计方式相同，而 $I_{34}$ 和 $I_{31}$ 的第二部分估计方式相同，所以

$\begin{array}{l} I_{32} \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2} \\ I_{34} \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (κ, ε_{1}) {‖ Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，对 $I_{33}$ 进行估计得到

$\begin{matrix} I_{33} = κ \int_{Ω} n (ϕ) A^{'} (ϕ) Δ ϕ q Δ^{2} ϕ d x \\ \leq κ c {‖ q ‖}_{L^{4}} {‖ Δ ϕ ‖}_{L^{4}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ Δ ϕ ‖}_{L^{2}}^{2}, \end{matrix}$

同理，对 $I_{35}$ 进行估计得到

$\begin{matrix} I_{35} = \int_{Ω} κ n (ϕ) A (ϕ) Δ q Δ^{2} ϕ d x \\ \leq κ c {‖ Δ q ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ \nabla q ‖}_{L^{2}}^{2} \end{matrix}$

将上述不等式相加得到 $I_{3}$ 的估计为

$I_{3} \leq ε {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{1} {‖ Δ q ‖}_{L^{2}}^{2} + ε_{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + c (κ, ε, ε_{1}, ε_{2}) ({‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2}) .$

令 $I_{4} = - \int_{Ω} \nabla (u \cdot \nabla q) \nabla q d x$ ，结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$\begin{matrix} I_{4} = \int_{Ω} \nabla (u \cdot \nabla q) \nabla q d x \\ \leq C {‖ u ‖}_{L^{4}} {‖ \nabla q ‖}_{L^{4}} {‖ Δ q ‖}_{L^{2}} \\ \leq ε {‖ Δ q ‖}_{L^{2}}^{2} + c (ε) {‖ u ‖}_{L^{2}}^{2} {‖ \nabla u ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2} . \end{matrix}$

令 $I_{5} = - \int_{Ω} \nabla (\frac{1}{τ (ϕ)} q) \nabla q d x$ ，同上可以得到

$I_{5} = \int_{Ω} \nabla (\frac{1}{τ (ϕ)} q) \nabla q d x \leq ε {‖ \nabla q ‖}_{L^{2}}^{2} + c (τ) {‖ \nabla ϕ ‖}_{L^{2}}^{2} {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ q ‖}_{L^{2}}^{2} .$

而对 $I_{6}$ 的估计可以写为

$\begin{matrix} I_{6} = \int_{Ω} \nabla (A (ϕ) Δ (A (ϕ) q)) \nabla q d x \\ = \int_{Ω} A (ϕ) A^{″} (ϕ) {(\nabla ϕ)}^{2} q Δ q d x + \int_{Ω} A (ϕ) A^{'} (ϕ) Δ ϕ q Δ q d x \\ + \int_{Ω} 2 A (ϕ) A^{'} (ϕ) \nabla ϕ \cdot \nabla q Δ q d x + \int_{Ω} {| A (ϕ) Δ q |}^{2} d x \\ = I_{61} + I_{62} + I_{63} + I_{64}, \end{matrix}$

对 $I_{61}, I_{62}, I_{63}$ 分别进行估计可以得到

$\begin{array}{l} I_{61} = \int_{Ω} A (ϕ) A^{″} (ϕ) {(\nabla ϕ)}^{2} q Δ q d x \leq ε {‖ A (ϕ) Δ q ‖}_{L^{2}}^{2} + c (ε) {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2}, \\ I_{62} = \int_{Ω} A (ϕ) A^{'} (ϕ) Δ ϕ q Δ q d x \leq ε {‖ Δ q ‖}_{L^{2}}^{2} + c (ε) {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} {‖ q ‖}_{L^{2}}^{2}, \\ I_{63} = \int_{Ω} 2 A (ϕ) A^{'} (ϕ) \nabla ϕ \cdot \nabla q Δ q d x \leq ε {‖ Δ q ‖}_{L^{2}}^{2} + c (ε) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2}, \end{array}$

将上述不等式相加可以得到

$I_{6} \leq ε_{1} {‖ A (ϕ) Δ q ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (ε_{1}, ε_{2}) ({‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} {‖ q ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla q ‖}_{L^{2}}^{2})$

令 $I_{7} = - \int_{Ω} \nabla (κ A (ϕ) d i v (n (ϕ) \nabla μ)) \nabla q d x$ ，

$\begin{matrix} I_{7} = κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q \nabla ϕ \cdot \nabla μ d x + κ \int_{Ω} A (ϕ) n (ϕ) Δ q Δ^{2} ϕ d x + κ \int_{Ω} A (ϕ) n (ϕ) Δ q Δ F^{'} (ϕ) d x \\ = I_{71} + I_{72} + I_{73}, \end{matrix}$

其中 $I_{71}$ 可以写成下式

$\begin{matrix} I_{71} = κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q \nabla ϕ \cdot \nabla μ d x \\ = κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q \nabla ϕ \cdot \nabla Δ ϕ d x + κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q {(\nabla ϕ)}^{2} {| ϕ |}^{2} d x, \end{matrix}$

对 $I_{71}$ 的第一部分进行估计可以得到

$\begin{array}{l} κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q \nabla ϕ \cdot \nabla Δ ϕ d x \\ \leq κ c {‖ Δ q ‖}_{L^{2}} {‖ \nabla ϕ ‖}_{L^{4}} {‖ \nabla Δ ϕ ‖}_{L^{4}} \\ \leq ε_{1} {‖ Δ q ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε_{1}, ε_{2}, κ) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

对 $I_{71}$ 的第二部分进行估计可以得到

$κ \int_{Ω} A (ϕ) n^{'} (ϕ) Δ q {(\nabla ϕ)}^{2} {| ϕ |}^{2} d x \leq c κ {‖ ϕ ‖}_{L^{\infty}}^{2} {‖ Δ q ‖}_{L^{2}} {‖ \nabla ϕ ‖}_{L^{4}}^{2} \leq ε {‖ Δ q ‖}_{L^{2}}^{2} + c (ε) {‖ Δ ϕ ‖}_{L^{2}}^{4},$

由于 $I_{72}$ 和 $I_{35}$ 的估计方式相同，所以

$\begin{matrix} I_{72} \leq \int_{Ω} κ n (ϕ) A (ϕ) Δ q Δ^{2} ϕ d x \\ \leq κ c {‖ Δ q ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \\ \leq ε_{1} {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + ε_{2} {‖ Δ q ‖}_{L^{2}}^{2} + c (κ, ε_{1}, ε_{2}) {‖ \nabla q ‖}_{L^{2}}^{2}, \end{matrix}$

同理 $I_{73}$ 可以也写成下式

$\begin{matrix} I_{73} = κ \int_{Ω} A (ϕ) n (ϕ) Δ q Δ F^{'} (ϕ) d x \\ = κ \int_{Ω} A (ϕ) n (ϕ) Δ q | ϕ | {(\nabla ϕ)}^{2} d x + κ \int_{Ω} A (ϕ) n (ϕ) Δ q {| ϕ |}^{2} Δ ϕ d x, \end{matrix}$

对 $I_{73}$ 的第一部分进行估计可以得到

$\begin{array}{l} κ \int_{Ω} A (ϕ) n (ϕ) Δ q | ϕ | {(\nabla ϕ)}^{2} d x \\ \leq κ c {‖ Δ q ‖}_{L^{2}} {‖ ϕ ‖}_{L^{\infty}} {‖ \nabla ϕ ‖}_{L^{4}}^{2} \\ \leq ε_{1} {‖ Δ q ‖}_{L^{2}}^{2} + ε_{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + c (ε_{1}, ε_{2}, κ) {‖ Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

对 $I_{73}$ 的第二部分进行估计可以得到

$κ \int_{Ω} A (ϕ) n (ϕ) Δ q {| ϕ |}^{2} Δ ϕ d x \leq κ c {‖ ϕ ‖}_{L^{\infty}}^{2} {‖ Δ ϕ ‖}_{L^{2}} {‖ Δ^{2} ϕ ‖}_{L^{2}} \leq ε {‖ Δ^{2} ϕ ‖}_{L^{2}}^{2} + c (ε) {‖ Δ ϕ ‖}_{L^{2}}^{4},$

令 $I_{8} = \int_{Ω} (u \cdot \nabla) u (- Δ u) d x$ ，结合Young’s不等式，Gagliardo-Nirenberg不等式，Hölder不等式，嵌入定理，得到

$I_{8} = \int_{Ω} (u \cdot \nabla) u \cdot (Δ u) d x \leq c {‖ u ‖}_{L^{4}} {‖ \nabla u ‖}_{L^{4}} {‖ Δ u ‖}_{L^{2}} \leq ε {‖ Δ u ‖}_{L^{2}}^{2} + c (ε) {‖ u ‖}_{L^{2}}^{2} {‖ \nabla u ‖}_{L^{2}}^{4} .$

令 $I_{9} = - \int_{Ω} η^{'} (ϕ) \nabla ϕ \cdot \nabla u Δ u d x$ ，同理可以得到

$I_{9} \leq c {‖ \nabla ϕ ‖}_{L^{4}} {‖ \nabla u ‖}_{L^{4}} {‖ Δ u ‖}_{L^{2}} \leq ε {‖ Δ u ‖}_{L^{2}}^{2} + c (ε) {‖ Δ ϕ ‖}_{L^{2}}^{2} {‖ \nabla u ‖}_{L^{2}}^{2} .$

令 $I_{10} = - \int_{Ω} \nabla ϕ μ Δ u d x$ ，同理可以得到

$I_{10} = \int_{Ω} \nabla ϕ F^{'} (ϕ) Δ u d x \leq c {‖ ϕ ‖}_{L^{\infty}}^{3} {‖ \nabla ϕ ‖}_{L^{2}} {‖ Δ u ‖}_{L^{2}} \leq ε {‖ Δ u ‖}_{L^{2}}^{2} + c (ε) {‖ \nabla ϕ ‖}_{L^{2}}^{4} {‖ \nabla u ‖}_{L^{2}}^{4} .$

再综合对 $I_{1} ~ I_{10}$ 的估计可以得到

$\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{Ω} ({| Δ ϕ |}^{2} + {| \nabla q |}^{2} + {| \nabla u |}^{2}) d x \\ \leq C_{1} ({‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla u ‖}_{L^{2}}^{2} + 1) ({‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla u ‖}_{L^{2}}^{2}) + C_{2} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2}, \end{array}$

由定理2.1的假设条件可知，对任意有限时间T

$\int_{0}^{T} ({‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla u ‖}_{L^{2}}^{2} + 1) d x \leq C,$

$\int_{0}^{T} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} d x \leq C,$

又因为(1.1)~(1.5)对任意有限时间T存在弱解，所以有

$\int_{0}^{T} {‖ \nabla ϕ ‖}_{L^{2}}^{2} d x \leq C, \int_{0}^{T} {‖ \nabla q ‖}_{L^{2}}^{2} d x \leq C, \int_{0}^{T} {‖ \nabla u ‖}_{L^{2}}^{2} d x \leq C,$

设 $A (t) = {‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla u ‖}_{L^{2}}^{2}, t \in (0, T)$ 且 $A (0) = {‖ Δ ϕ_{0} ‖}_{L^{2}}^{2} + {‖ \nabla q_{0} ‖}_{L^{2}}^{2} + {‖ \nabla u_{0} ‖}_{L^{2}}^{2}$ ，

利用引理2.2 (Gronwall不等式)，带入对 $I_{1} ~ I_{10}$ 的估计可以得到

$\begin{array}{l} \sup A (t) \\ \leq (A (0) + \int_{0}^{T} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} d x) \exp {T + \int_{0}^{T} {‖ \nabla Δ ϕ ‖}_{L^{2}}^{2} + {‖ Δ ϕ ‖}_{L^{2}}^{2} + {‖ \nabla ϕ ‖}_{L^{2}}^{2} + {‖ \nabla q ‖}_{L^{2}}^{2} + {‖ \nabla u ‖}_{L^{2}}^{2} d x} \\ < \infty \end{array}$

综上所述，可以证明定理2.1成立，那么由弱解的存在性，可以得到粘弹性相分离模型在二维空间中整体强解的存在性。

文章引用

裴小田. 粘弹性相分离模型在二维空间中强解的整体存在性
The Overall Existence of Strong Solution for Viscoelastic Phase Separation Model in the Two Dimensional Space[J]. 应用数学进展, 2023, 12(06): 2749-2757. https://doi.org/10.12677/AAM.2023.126275

参考文献

1. Brunk, A. and Lukacova-Medvidova, M. (2022) Global Existence of Weak Solutions to Viscoelastic Phase Separation: Part I Regular Case. arXiv:1907.03480v3.

2. Lukacova-Medvidova, M., Mizerova, H. and Necasova, S. (2015) Global Existence and Uniqueness Result for the Diffusive Peterlin Viscoelastic Model. Nonlinear Analysis: Theory, Methods & Applications, 120, 154-170. https://doi.org/10.1016/j.na.2015.03.001

3. Strasser, P.J., Tierra, G., Dunweg, B. and Lukacova-Medvidova, M. (2019) Energy-Stable Linearschemes for Polymer-Solvent Phase Field Models. Computers & Mathematics with Applica-tions, 77, 125-143.

4. Zhou, D., Zhang, P.W. and E, W.N. (2006) Modified Models of Polymer Phase Separation. Physical Review E, 73, Article ID: 061801. https://doi.org/10.1103/PhysRevE.73.061801

5. Chupin, L. (2003) Existence Result for a Mixture of Non Newtonian Flows with Stress Diffusion Using the Cahn- Hilliard Formulation. Discrete and Continuous Dynamical Systems-B, 3, 45-68. https://doi.org/10.3934/dcdsb.2003.3.45

6. Abels, H. (2009) On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities. Archive for Rational Mechanics and Analysis, 194, 463-506. https://doi.org/10.1007/s00205-008-0160-2

7. Abels, H., Depner, D. and Garcke, H. (2013) On an Incompressi-ble Navier-Stokes/Cahn-Hilliard System with Degenerate Mobility. Annales de l’Institut Henri Poincaré Analyse Nom Linéaire, 30, 1175-1190. https://doi.org/10.1016/j.anihpc.2013.01.002

8. Spiller, D., Brunk, A., Habrich, O., Egger, H., Lukaco-va-Medvidova, M. and Dunweg, B. (2021) Systematic Derivation of Hydrodynamic Equations for Viscoelastic Phase Separation. Journal of Physics: Condensed Matter, 33, Article ID: 364001. https://doi.org/10.1088/1361-648X/ac0d17

9. Tanaka, H. (2000) Viscoelastic Phase Separation. Journal of Physics: Condensed Matter, 12, R207. https://doi.org/10.1088/0953-8984/12/15/201

10. Lukacova-Medvidova, M., Mizerova, H., Notsu, H. and Tabata, M. (2017) Numerical Analysis of the Oseen-Type Peterlin Viscoelastic Model by the Stabilized Lagrange-Galerkin Method. Part II: A Linear Scheme. ESAIM: M2AN, 51, 1663-1689. https://doi.org/10.1051/m2an/2017032

期刊菜单