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Computer-assisted equilibrium validation for the diblock copolymer model

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  • The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.

    Mathematics Subject Classification: Primary:35B40, 35B41;Secondary:35K55, 60F10, 60H15, 74N99.

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  • Figure 1.  Numerically computed bifurcation diagrams for the diblock copolymer model (2) on the domain $\Omega = (0,1)$ and with $\sigma = 6$. The left diagram is for total mass $\mu = 0$, while the right diagram is for $\mu = 0.3$. The solution measure for both diagrams is the $L^2(0,1)$-norm of the solutions. The solution branches are color-coded by the numerically determined Morse index of the solutions, and the colors black, red, blue, green, magenta, and cyan correspond to indices $0$, $1$, $2$, $3$, $4$, and $5$, respectively. The bifurcation parameter is $\lambda$.

    Figure 2.  Numerically computed bifurcation diagrams for the diblock copolymer model (2) on the domain $\Omega = (0,1)$ and with $\sigma = 6$. The left diagram is for total mass $\mu = 0$, while the right diagram is for $\mu = 0.3$. The solution measure for both diagrams is the energy of the solutions as defined in (1. The solution branches are color-coded by the numerically determined Morse index of the solutions, and the colors black, red, blue, green, magenta, and cyan correspond to indices $0$, $1$, $2$, $3$, $4$, and $5$, respectively. The bifurcation parameter is $\lambda$.

    Figure 3.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0$ and $\sigma = 6$. The left diagram shows solutions for $\lambda = 100$, while the right diagram is for $\lambda = 150$. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.

    Figure 4.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0$ and $\sigma = 6$. Both diagrams shows solutions for $\lambda = 200$. The ones on the left correspond to the solutions which are validated in Table 1 below, while the two solutions on the right are validated in Table 2. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, blue functions have index $2$, and the green solution has index $3$.

    Figure 5.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0.3$ and $\sigma = 6$. The left diagram shows solutions for $\lambda = 100$, while the right diagram is for $\lambda = 200$. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.

    Figure 6.  Choice of the constant $\delta_\lambda$ in Theorem 2.2. In both diagrams, the blue lines indicate the identities $\delta_\lambda = (\delta_u - \delta_1) / (2 M_3 K)$ and $\delta_\lambda = M_1 (\delta_3 - \delta_u) / M_2$, which form the upper boundary of the region of admissible pairs $(\delta_u, \delta_\lambda)$. The left diagram depicts the normal situation, in which $\delta_3<d_1$. In the right image we illustrate the case $\delta_1<d_1<\delta_3$, in which the admissible region is reduced due to hypothesis (H3). For both diagrams, we assume that $\delta_{\lambda,opt}<d_2 \le d_3$, which is usually satisfied.

    Table 1.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For the $\lambda$-values $100$, $150$, and $200$ we consider representative solutions from most of the branches shown in the left diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the index of each solution, the value $\varrho $ of the residual in (H1), as well as the values $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. All computations use Proposition 3.5 with $K = 100$.

    Equilibrium Validation for $\mu = 0$ with $K = 100$
    $\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
    100 #1 0 2.020897e-14 2.020897e-12 7.512488e-06 190
    #2 0 2.222804e-14 2.222804e-12 1.547854e-05 106
    #3 1 4.444820e-14 4.444820e-12 1.394691e-05 112
    #4 2 3.276268e-15 3.276268e-13 1.625587e-05 59
    150 #1 0 3.632609e-14 3.632611e-12 4.268987e-06 332
    #2 1 4.705001e-14 4.705003e-12 8.092680e-06 584
    #3 1 1.814983e-14 1.814984e-12 3.929213e-06 508
    #4 2 2.672918e-14 2.672918e-12 5.189605e-06 232
    #5 2 2.310413e-14 2.310413e-12 7.824295e-06 140
    200 #1 0 4.799846e-14 4.799850e-12 2.932214e-06 479
    #4 1 6.153889e-14 6.153893e-12 5.080967e-06 314
    #5 2 6.674118e-14 6.674124e-12 3.937197e-06 301
    #6 2 3.750452e-14 3.750453e-12 6.304030e-06 142
    #7 3 7.652149e-15 7.652150e-13 3.130001e-06 676
     | Show Table
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    Table 2.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For $\lambda = 200$ we consider representative solutions from the second and third branches shown in the left diagram of Figure 1, numbered from top to bottom. The table lists the index of each solution, the value $\varrho $ in (H1), as well as $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. The computations use $K = 500$.

    Equilibrium Validation for $\mu = 0$ with $K = 500$
    $\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
    200 #2 0 2.642564e-14 1.321300e-11 4.858832e-07 1828
    #3 1 6.084200e-14 3.042193e-11 4.959956e-07 1108
     | Show Table
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    Table 3.  Rigorous numerical results for the diblock copolymer model with $\mu = 0.3$. For the $\lambda$-values $100$ and $200$ we consider representative solutions from each of the branches shown in the right diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the index of each solution, the value $\varrho $ of the residual in (H1), as well as the values $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. All computations use Proposition 3.5 with $K = 100$.

    Equilibrium Validation for $\mu = 0.3$ with $K = 100$
    $\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
    100 #1 0 5.728511e-14 5.728512e-12 1.047050e-05 106
    #2 0 2.129870e-14 2.129870e-12 8.569644e-06 175
    #3 1 2.129870e-14 2.129870e-12 8.569644e-06 811
    #4 1 6.466218e-15 6.466218e-13 4.747448e-05 64
    200 #1 0 8.153471e-14 8.153483e-12 2.813889e-06 644
    #2 1 7.326258e-14 7.326268e-12 2.834757e-06 552
    #3 1 1.075251e-13 1.075252e-11 4.071324e-06 180
    #4 2 8.642886e-14 8.642896e-12 3.858723e-06 271
    #5 2 2.696057e-14 2.696057e-12 1.384417e-05 113
     | Show Table
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    Table 4.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For the $\lambda$-values $100$ and $150$ we consider representative solutions from each of the branches shown in the left diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the values of $\delta_1$, $\delta_2$, $\delta_3$, and $\delta_{\lambda,opt}$ from Theorem 2.2. The last two columns indicate a numerical estimate $K_{est}$ for the optimal constant $K$ in (H2), as well as the value of $K$ used for the validation computation.

    Branch Validation for $\mu = 0.0$ with estimated $K$
    $\lambda$ No. $\delta_1$ $\delta_2$ $\delta_3$ $\delta_{\lambda,opt}$ $K_{est}$ $K$
    100 #1 1.0446e-13 1.1065e-04 1.1079e-04 2.6068e-04 2.2 3.3
    #2 5.1001e-13 8.9555e-05 8.9572e-05 2.6200e-05 5.7 8.6
    #3 3.0910e-13 1.7520e-04 1.7527e-04 1.2525e-04 2.6 3.9
    #4 8.1585e-14 5.3658e-05 5.3726e-05 9.5155e-05 10.0 15.1
    150 #1 2.1895e-13 5.7574e-05 5.7619e-05 1.3199e-04 2.4 3.7
    #2 5.7821e-12 8.6076e-06 8.6078e-06 4.4425e-07 31.3 46.9
    #3 3.3317e-13 1.7562e-05 1.7569e-05 2.1107e-05 7.4 11.1
    #4 1.5386e-13 9.3999e-05 9.4056e-05 1.5404e-04 1.8 2.7
    #5 1.3800e-13 1.3328e-04 1.3336e-04 1.8173e-04 1.9 2.9
     | Show Table
    DownLoad: CSV
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