Constructions of various solutions for parabolic equations in mathematical biology and phase-field models

概要

In this dissertation we consider well-posedness and constructions of various solu- tions for the parabolic evolution equations by theories of analytic semigroups and max- imal Lp regularity. The equations we consider are the bidomain equations which repre- sent electrophysiological wave propagation in the heart, the phase-field Navier–Stokes equations which represent the deformation of the vesicle membrane in incompressible viscous fluids, the Cahn–Hilliard equation which represent the spinodal decomposi- tion of binary mixture, and the abstract parabolic evolution equations. The bidomain equations are the model of biology and the phase-field Navier–Stokes equations and the Cahn–Hilliard equations are so-called phase-field models.

In Chapter 1 we consider the semigroup generated by the principal part of the bido- main equations. In general, solutions of the linear evolution equations can be analyzed by the semigroup which is the solution operator corresponding from the initial data to the solution at some time. It is important to characterize whether the semigroup is analytic or not. Analytic semigroups represent the smoothing effect in parabolic evolu- tion equations. The bidomain equations is complicated since they have three unknown functions ui,e and u. Bourgault et al. introduced a bidomain operator and they re- garded the equation into a reaction diffusion system ([Bourgault et al. 2009]). They proved that the bidomain operator is a self-adjoint operator and a non-negative opera- tor in L2 space. It means that the operator A generates an analytic semigroup e−tA . In this chapter, we consider the bidomain operator in Lp spaces for 1 < p ≤ ∞. We prove an L ∞ resolvent estimate by a contradiction argument and a blow-up argument. From the inequality obtained by the negation of its conclusion, we show that one holds the inequality by compactness, but the other breaks down the inequality by uniqueness, which leads a contradiction. The estimates from L2 and L∞ imply an Lp resolvent es- timate for 1 < p < ∞ by the interpolation and the duality. We properly introduce the bidomain operator in Lp spaces and characterize the resolvent set of the operator. The main theorem is that the bidomain operator generates an analytic semigroup in L p spaces. For the non-linear bidomain equations, we construct a local well-posedness theorem by a general theory of analytic semigroups. This chapter consists of a joint work with Professor Giga (1).

In Chapter 3 we continue discussing the time periodic problem about the bido- main equations. In Chapter 2 we need the smallness conditions since the bidomain equations are considered as a perturbed equation of their linearized equations. In this chapter we restrict that the nonlinear term is FitzHugh–Nagumo type and the function space is L2 , but we prove that the equations admit a time periodic solution without assuming the smallness conditions on the source. When we use Galerkin method, we apply Brouwer ’s fixed point theorem for Poincare map. This guarantees the existence ´ of a weak time periodic solution. Moreover we regard the initial data as this periodic solution, then by global well-posedness result for initial value problem, the weak solu- tions agree with the strong solutions. This is a regularity theorem, which implies the existence of the strong time periodic solutions in maximal L2 -L2 spaces. This chapter consists of a joint work with Professor Giga and Mr. Kress (3).

In Chapter 4 we consider the well-posedness for the phase-field Navier–Stokes equations. In previous works, it was proved the existence of the global weak solutions ([Du et al. 2007]) and the unique existence of the local strong solutions ([Takahashi et al. 2012]). However the former is not known its uniqueness and regularity and the latter is analyzed as a semi-linear evolution equation although the coupling part should be the principal part. Therefore its regularity class of the solutions is not suitable. We treat the equation as a quasi-linear evolution equation which means the coupling part is the principal part. We prove the linear operator has maximal Lp -Lq regularity property and prove the unique existence of the local strong solutions and the continuity on the initial data. Moreover we have that the solution is real analytic in time and space, thus this is a classical solution. At last it is shown that the variational strict stable solution is exponentially stable provided the product of the viscosity coefficient and the mobility constant is large. This chapter consists of a paper (4).

In Chapter 5 we consider the global existence and uniqueness of the solutions for the Cahn–Hilliard equation. Let the order parameter u and the chemical potential µ. In previous works, the boundary condition was Neumann condition for µ so that it derives the volume conservation law dtd ´Ω u dx = 0. Since the equation is fourth order, we need two boundary conditions. The other boundary condition is Neumann con- dition for u so that it derives that the energy EΩ (u) decrease, or a dynamic boundary condition for u so that it derives the energy EΩ (u) + E∂ Ω (u) decrease. Here E∂ Ω (u) is a energy from the boundary. However when the substances permeate the boundary, volume preservation is not necessarily achieved. Gal and Goldstein et al. introduced new boundary conditions which model the boundaries are permeable walls and non- permeable walls, respectively ([Gal 2006], [Goldstein et al. 2011]). The former derives d dt ( ´ Ωu dx + ´ ∂ Ω u dS ) = −c ´ Γ µ dSb with the constants b > 0, c ≥ 0 and the latter derives the total volume conservation law dtd (´Ω u dx + ´∂ Ω u dS ) = 0. In permeable walls, the case c = 0 implies the total volume conservation law. We consider these two boundary conditions including the case c = 0. We apply the linear theory of maximal Lp regular- ity which corresponds to higher order equations and the dynamic boundary condition (cf. [Denk et al. 2008]). It characterizes the solvability and the classes of data by a nec- essary and a sufficient condition. For the non-linear Cahn–Hilliard equation, we use Banach’s fixed point theorem, energy estimates and a-priori estimate. We are able to generalize the global solvability for p ̸= 2 although above previous works are L2 frame- work. Moreover since the spaces of initial functions are optimal, we are able to classify the necessary of the compatibility conditions by p although above previous works need the compatibility conditions. This chapter consists of a paper (5).

In Chapter 6 we consider the well-posedness for the abstract parabolic evolution equations by means of maximal Lp regularity with time weights. The (classical) max- imal Lp regularity property is the solvability and the estimate for the abstract linear evolution equations u′ + Au = f in Lp (0, T ; X ). The remarkable application of the maximal Lp regularity is the solvability of the quasi-linear parabolic evolution equa- tions u′ + A(u)u = F (u). However when we use this theory, in general, we need to take the initial data in the real interpolation space (X, D(A))1−1/p,p which is the trace space of the solution space at t = 0. The theory of maximal Lp regularity with time weight is a generalization of this initial data while keeping the solution class except- ing for the behavior near t = 0. In a series of papers by J. Pruss, they extended the ¨ initial data to (X, D(A))µ−1/p,p for µc ≤ µ ≤ 1, where the critical weight µc is deter- mined by the non-linearities F . Moreover they give a sufficient condition to be a global solution when the equation is a semi-linear and the non-linear term is the bi-linear. In this chapter we extend this general local well-posedness theory to the quasi-linear parabolic evolution equations u′ + A(t, u) = F (t, u) with the time-dependent operator and the non-linear term. Under the case A(t, u) = A(t) and the assumptions used in the local well-posedness with some technical assumption, we give a sufficient condition in order that the local solution becomes a global solution. This is a generalization of above bilinear non-linearities. As a example of the local well-posedness theorem, we applied the theory to the quasi-linear heat equations u′ − a(t, u)∆u = F (t, u) + |∇u|κ with κ > 2.

All chapters are based on the papers below, respectively. All sections, notations and theorems, etc are cited only in each chapter where they appear.