Special Solutions to a Nonlinear Coarsening Model with Local Interactions

Abstract

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the backward parabolic equation \(\partial _t x = - \frac{\beta }{|\beta |} \Delta x^\beta \), with \(\beta \) in the fast diffusion regime \((-\infty ,0) \cup (0,1]\). Sites with mass zero are deleted from the system, which leads to a coarsening of the mass distribution. The rate of coarsening suggested by scaling is \(t^\frac{1}{1-\beta }\) if \(\beta \ne 1\) and exponential if \(\beta = 1\). We prove that such solutions actually exist by an analysis of the time-reversed evolution. In particular we establish positivity estimates and long-time equilibrium properties for discrete parabolic equations with initial data in \(\ell _+^\infty (\mathbb {Z})\).

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Acknowledgements

The author would like to thank Barbara Niethammer and Juan J. L. Velázquez for inspiration, helpful discussions and proofreading. This work was supported by the German Research Foundation through the CRC 1060 The Mathematics of Emergent Effects.

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Correspondence to Constantin Eichenberg.

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Communicated by Paul Newton.

A Appendix

A Appendix

Here we address all technical results that were used in the previous sections and either prove them or give a reference. In the first three parts we discuss aspects of the discrete fast diffusion equation, while the rest of the appendix contains results about parabolic Hölder regularity in the discrete setting.

A.1 The Equation \(\partial _t u = \Delta G_\beta (u)\)

We consider the Cauchy problem for the discrete fast diffusion equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u = \frac{\beta }{|\beta |} \Delta u^\beta = \Delta G_\beta (u) \quad \mathrm {in} \ (0,\infty ) \times \mathbb {Z}, \\ u(0,\cdot ) = u_0, \end{array}\right. } \end{aligned}$$
(A.1)

with \(\beta \in (-\infty ,0) \cup (0,1]\), \(u_0 \in \ell _+^\infty (\mathbb {Z})\) and

$$\begin{aligned} \Delta u = u(k-1) -2u(k) + u(k+1). \end{aligned}$$
(A.2)

We are concerned with the long-time existence of classical solutions:

Definition A.1

A function \(u: [0,\infty ) \rightarrow \ell _+^\infty (\mathbb {Z})\) is a solution to problem (A.1) if the following conditions are satisfied:

  1. 1.

    \(t \mapsto u(t,\cdot )\) is in \(C^0([0,\infty );\ell _+^\infty (\mathbb {Z}))\) and \(u(0,\cdot ) = u_0\).

  2. 2.

    For every \(k \in \mathbb {Z}\) we have \(u(.,k) \in C^1((0,\infty );\mathbb {R}_{> 0})\) and

    $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} u(t,k) = \Delta G_\beta (u)(t,k), \end{aligned}$$
    (A.3)

    for all \(k \in \mathbb {Z}\) and \(t > 0\).

For positive \(\beta \) it is not hard to prove the existence of a solution for any kind of initial data, since \(G_\beta \) is bounded at zero and one has simple a priori estimates due to the comparison principle (see below). For negative \(\beta \) the existence of a sufficiently regular solution for arbitrary data cannot be expected due to \(G_\beta (x)\) becoming singular at \(x = 0\). Similar to the result in Helmers et al. (2016), we have to restrict to initial data which satisfy a certain positivity condition:

Definition A.2

For \(u \in \ell _+^\infty (\mathbb {Z})\) and \(d > 0\) let

$$\begin{aligned} \sigma _+(u,k,d)&= \inf \{l > k: u(l) \ge d \}. \end{aligned}$$
(A.4)

Then for any \(L \in \mathbb {N}\) and \(d > 0\) we define

$$\begin{aligned} \mathcal {P}_{L,d} = \left\{ u \in \ell _+^\infty : \ \sup _{k \in \mathbb {Z}} \sigma _+(u,k,d) \le L \right\} . \end{aligned}$$
(A.5)

In other words, \(u \in \mathcal {P}_{L,d}\) means that particles with large mass cannot be very far apart. This is also relevant for the case \( \beta > 0\) since it allows us to prove certain Harnack-type positivity estimates. We have the following result:

Theorem A.3

Let \(\beta \in (-\infty ,0) \cup (0,1]\), and consider initial data \(u_0 \in \mathcal {P}_{L,d}\). Then the following statements hold:

  1. 1.

    If \(\beta \in (-\infty ,0) \cup (0,1)\), there exists a positive constant \(c = c \left( L, d, \beta , ||u_0||_\infty \right) \) and a solution u to Eq. (A.1) on \([0,\infty )\) with initial data \(u_0\) satisfying

    $$\begin{aligned} u(t,k)&\ge c \left( 1 \wedge t^{\frac{1}{1-\beta }} \right) , \ \mathrm {for} \ \mathrm {all} \ k \in \mathbb {Z}. \end{aligned}$$
    (A.6)
  2. 2.

    If \(\beta = 1\), the same statement holds with estimate (A.6) replaced by

    $$\begin{aligned} u(t,k)&\ge c \exp (-2t)I_L(2t), \end{aligned}$$
    (A.7)

    where \(I_L(t)\) denotes the L-th modified Bessel function of the first kind.

  3. 3.

    Comparison principle: If \( c_1 \le u_0 \le c_2\), then u satisfies these bounds for all times.

In this and the next two sections we give a full proof of the above result. The general strategy to prove existence of solutions for Eq. (2.21) is to use regularisation and standard ODE theory. Instead of infinitely many particles with nonnegative mass we first consider a periodic N-particle ensemble where particles have strictly positive mass. The first important a priori estimate is the comparison principle:

Lemma A.4

(Finite positive ensemble) Let \(\mathbb {T}_N\) denote the one-dimensional periodic lattice with N lattice points. Let \(u_0 \in \ell _+^\infty (\mathbb {T}_N)\) with \( 0 < \delta \le u_0 \le C\). Then there exists a unique solution \(u:[0,\infty ) \rightarrow \ell _+^\infty (\mathbb {T}_N)\) of (A.1) with \(\delta \le u(t,\cdot ) \le C\).

Proof

The proof is very similar to Lemma 2 in Esedoglu and Greer (2009) and a standard maximum principle argument. Because \(u_0 \ge \delta \), standard ODE theory gives the existence and uniqueness of a smooth solution u on the time interval \([0,t^*]\) to Eq. (A.1) with \(\delta /2 \le u(t,.) \le 2C\) for some positive \(t^*\). For small \(\varepsilon > 0\) we then consider the solution \(u_\varepsilon \) of the modified problem

$$\begin{aligned} \partial _t u_\varepsilon&= \Delta G_\beta (u_\varepsilon ) + \varepsilon , \end{aligned}$$
(A.8)
$$\begin{aligned} u_\varepsilon (0,\cdot )&= u_0, \end{aligned}$$
(A.9)

that exists on the same time interval as u and satisfies the same bounds after possibly making \(t^*\) smaller. Because \(G_\beta \) is smooth on \([\delta /2,2C]\) we have that \(u_\varepsilon \rightarrow u\) uniformly on \([0,t^*]\). We claim that \(u_\varepsilon \) attains its minimum over \([0,t^*] \times \mathbb {T}_N\) at \(t = 0\). If not, there exists \(t_0 \in (0,t^*]\) and \(k_1\) such that \(u_\varepsilon (t_1,k_1)\) is the absolute minimum. Consequently we get

$$\begin{aligned} 0 \ge \partial _t u_\varepsilon (t_1,k_1) = \Delta G_\beta (u_\varepsilon )(t_1,k_1) + \varepsilon \ge \varepsilon , \end{aligned}$$
(A.10)

a contradiction. Here we used that \(G_\beta \) is increasing. Hence \(u_\varepsilon (t,.) \ge \delta \) for all \(t \in [0,t^*]\) and, sending \(\varepsilon \rightarrow 0\), the same bound holds for u. The same argument for the maximum where \(+\varepsilon \) is replaced with \(-\varepsilon \) yields that \(u \le C\). Iterating from \(t = t^*\), we see that the solution can be extended to \([0,\infty )\) and always satisfies the desired bounds. \(\square \)

From this result we can easily pass to the limit as \(N \rightarrow \infty \) to obtain solutions for infinite numbers of particles:

Corollary A.5

(Infinite positive ensemble) Let \(u_0 \in \ell _+^\infty (\mathbb {Z})\) with \( 0 < \delta \le u_0 \le C\). Then there exists a solution \(u:[0,\infty ) \rightarrow \ell _+^\infty (\mathbb {Z})\) of (A.1) with \(\delta \le u(t,.) \le C\).

Proof

This is a standard compactness argument. We choose \(u_0^{(N)}\) to be N-periodic such that \(u_0^{(N)}(k) \rightarrow u_0(k)\) for each \(k \in \mathbb {Z}\). Let \(u^{(N)}\) be the corresponding solutions from the above lemma. Then due to the a priori bounds \(\delta \le u^{(N)} \le C\) and Eq. (A.1) we have

$$\begin{aligned} \left| \frac{\mathrm {d}}{\mathrm {d}t} u^{(N)} \right| = \left| \Delta G_\beta \left( u^{(N)} \right) \right| \le K(\delta ,C), \end{aligned}$$
(A.11)

hence the solutions are uniformly Lipschitz continuous. Applying the Arzela-Ascoli Theorem and a diagonal argument we can extract a convergent subsequence (not relabelled) such that \(u^{(N)}(\cdot ,k) \rightarrow u(\cdot ,k)\) uniformly on compact time intervals, where \(u(\cdot ,k) \in C^0([0,\infty )\). In particular u satisfies the same bounds as \(u^{(N)}\). Integrating (A.1) in time and passing to the limit (which is possible due to the a priori bounds) then yields

$$\begin{aligned} u(t,k) = u_0(k) + \int _{0}^{t} \Delta _\sigma G_\beta (u)(s,k) \ \mathrm {d}s. \end{aligned}$$
(A.12)

This in turn shows that u(., k) is continuously differentiable and solves (A.1) pointwise. Again, the bounds on u yield Lipschitz continuity in \(\ell _+^\infty (\mathbb {Z})\). \(\square \)

For our purpose, we need the existence of solutions in particular for initial data with mass-zero particles. The general strategy is to approximate the initial data by regularised data via

$$\begin{aligned} u_{0,\delta } = u_0 \vee \delta . \end{aligned}$$
(A.13)

The above existence result then yields long-time solutions \(u_\delta \) with initial data \(u_{0,\delta }\). In the case \(\beta > 0\) one can pass to the limit \(\delta \rightarrow 0\) in the same manner as above, since \(G_\beta \) is bounded at zero, yielding a general existence result:

Corollary A.6

(Existence for positive \(\beta \)) Let \(\beta \in (0,1]\) and \(u_0 \in \ell _+^\infty (\mathbb {Z})\). Then there exists a solution \(u:[0,\infty ) \rightarrow \ell _+^\infty (\mathbb {Z})\) of (A.1).

Alternatively it is likely possible to prove this result directly via an infinite dimensional fixed-point method. Since we need Corollary (A.5) for the case \(\beta < 0\) anyway, the above method is the fastest for our purpose. In the next section we deal with the negative \(\beta \) case, including existence and the positivity estimate (A.6). Then we prove the positivity estimate for positive \(\beta \), completing the proof of Theorem A.3.

A.2 Existence of Solutions for \(\beta < 0\)

In the following we always assume \(\beta < 0\). The key idea to prove existence of solutions is to exploit the fact that regions which are enclosed by large particles (called traps) are screened from the rest of the particles, very similar to Helmers et al. (2016). One important difference however is the fact that the backward equation does not yield a priori estimates on the persistence of traps. We make the following definition:

Definition A.7

We say that a solution u to Eq. (A.1) with initial data \(u_0 \in \mathcal {P}_{L,d}\) has the persistence property on [0, T] if \(u(0,k) \ge d\) implies \(u(t,k) \ge \frac{d}{2}\) for all \(t \in [0,T]\).

By making use of the theory for the coarsening equation developed in Helmers et al. (2016) we have the following result concerning Hölder regularity:

Lemma A.8

There exist constants \(T' = T'(\beta ,d)\) and \(C = C(\beta ,L) > 0\) such that the following holds: If a solution u to Eq. (A.1) with initial data \(u_0 \in \mathcal {P}_{L,d}\) has the persistence property on [0, T] and \(T \le T'\), then

$$\begin{aligned} |u(t_2,k)-u(t_1,k)|&\le C|t_2 - t_1|^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.14)

for all \(t_2,t_1 \in [0,T]\) and \(k \in \mathbb {Z}\).

Proof

We consider the time-reversed function

$$\begin{aligned} x(s,k) = u(T - s,k), \end{aligned}$$
(A.15)

then x is a solution to the coarsening equation for \( 0 \le s \le T\) that satisfies \(x(0,.) \in \mathcal {P}_{L,\frac{d}{2}}\). Applying Lemma 3.3 from Helmers et al. (2016) (if \(T \le T^*(\beta ,\frac{d}{2}) =: T'\)) yields the desired Hölder continuity for x, and thus also for u.

From this result we derive the first a priori estimate:

Lemma A.9

Let \(u_{0,\delta }\) be as above and let \(u_\delta \) be the corresponding solution of Eq. (A.1), which exists by Lemma A.5. Then there exists \(T = T(L,d) > 0\) such that \(u_\delta \) has the persistence property on [0, T].

Proof

First we note that because the solution satisfies \(u_\delta \ge \delta \) for all times we have the Lipschitz estimate

$$\begin{aligned} |\partial _t u_\delta | \le 4\delta ^{\beta }. \end{aligned}$$
(A.16)

This means that \(u_\delta (0,k) \ge d\) implies \(u_\delta (t,k) \ge \frac{d}{2}\) for \(0 \le t \le t_0\), where

$$\begin{aligned} t_0 = \frac{d}{8\delta ^\beta }. \end{aligned}$$
(A.17)

Let T be the largest time such that \(u_\delta (0,k) \ge d\) implies \(u_\delta (t,k) \ge \frac{d}{2}\) on [0, T]. By the above considerations we already know that \(T > 0\). If \(T \le T'(\beta ,d)\) we can apply Lemma A.8 and get

$$\begin{aligned} u_\delta (t,k) \ge u_\delta (0,k) - Ct^{\frac{1}{1-\beta }}. \end{aligned}$$
(A.18)

If \(u_\delta (0,k) \ge d\), this implies \(u_\delta (T,k) \ge d - CT^\frac{1}{1-\beta }\). On the other hand, by the definition of T there exists such a k with \(u_\delta (T,k) \le \frac{3d}{4}\), hence

$$\begin{aligned} \frac{3d}{4} \ge d - C(L)T^\frac{1}{1-\beta }, \end{aligned}$$
(A.19)

which gives a lower bound for T in terms of L and d. \(\square \)

The next a priori estimate is crucial to get uniform Hölder bounds on \(u_\delta \), as well as integral bounds which are needed to pass to the limit.

Lemma A.10

Let \(u_\delta \) be as above. Then there exists \(c = c(\beta ,L,d) > 0\) and \(t^* = t^*(\beta ,L,d) > 0\) such that

$$\begin{aligned} u_\delta (t,\cdot ) \ge ct^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.20)

for \(0 \le t \le t^*\).

Proof

We apply a very similar argument as in the proof of Lemma 3.5 in Helmers et al. (2016). If the statement is false, there exist sequences \(u^{(n)}\), \(t_n \rightarrow 0\), \(\delta _n \rightarrow 0\) and \(k_n \in \mathbb {Z}\) such that

$$\begin{aligned} u^{(n)}_{\delta _n}(t_n,k_n) \le \frac{1}{n} t_n^{\frac{1}{1-\beta }}. \end{aligned}$$
(A.21)

By translation invariance we can assume that \(k_n = k_0\) is constant. We rescale and define

$$\begin{aligned} v_n(s,k) = t_n^{\frac{1}{\beta -1}}u^{(n)}_{\delta _n}(t_ns,k). \end{aligned}$$
(A.22)

Then \(v_n\) is a solution to Eq. (A.1) with \(v_n(1,k_0) \rightarrow 0\). Additionally we have \(v_n(0,.) \in \mathcal {P}_{L,d}\) and \(v_n\) satisfies the persistence property on [0, 1] for large n by Lemma A.9. Since \(t_n^{\frac{1}{\beta -1}} d \rightarrow \infty \) and \(T' \rightarrow \infty \) as \(d \rightarrow \infty \) we also have that \(v_n\) is uniformly Hölder continuous by Lemma A.8. Let B be the largest set of consecutive indices containing \(k_0\) such that

$$\begin{aligned} \liminf _{n \rightarrow \infty } v_n(1,k) = 0, \end{aligned}$$
(A.23)

for \(k \in B\). Observe that we have \(|B| \le L\) due to \(t_n^{\frac{1}{\beta -1}} d \rightarrow \infty \) and the persistence property. Let \(l_-, l_+\) be the nearest particle index to the left, respectively, to the right of B. We restrict to a subsequence such that \(v_n(1,k) \rightarrow 0\) for \(k \in B\) and \(v_n(1,l_\pm ) \ge \lambda > 0\). If we define the local mass \(M_n(s)\) as

$$\begin{aligned} M_n(s) = \sum _{k \in B}v_n(s,k), \end{aligned}$$
(A.24)

then an elementary calculation gives

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}M_n(s) = v_n^\beta (s,l_- + 1) - v_n^\beta (s,l_-) + v_n^\beta (s,l_+ -1) - v_n^\beta (s,l_+). \end{aligned}$$
(A.25)

Due to uniform Hölder continuity and particles in B going to zero there exists \(\varepsilon > 0\) such that

$$\begin{aligned} v_n^\beta (s,l_\pm ) \le \frac{1}{2} v_n^\beta (s,l_\pm \mp 1), \end{aligned}$$
(A.26)

for \(s \in [1-\varepsilon ,1]\) and large enough n. Using Eq. (A.25) on this time interval we obtain

$$\begin{aligned} 2\frac{\mathrm {d}}{\mathrm {d}s}M_n(s) \ge v_n^\beta (s,l_- + 1) + v_n^\beta (s,l_+ - 1) \ge M_n^{\beta }(s), \end{aligned}$$
(A.27)

which, after integrating from \(1-\varepsilon \) to 1 yields

$$\begin{aligned} M_n(1) \ge \tilde{\varepsilon } > 0, \end{aligned}$$
(A.28)

which gives a contradiction after sending n to infinity. \(\square \)

This result gives us the a priori estimates we need to pass to the limit:

Corollary A.11

(Hölder continuity) Let \(u_\delta \) and \(t^*\) be as above. Then there exists \(C = C(\beta ,L,d)\) such that

$$\begin{aligned} |u_\delta (t_2,k)-u_\delta (t_1,k)|&\le C|t_2 - t_1|^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.29)

for all \(t_2,t_1 \in [0,t^*]\) and \(k \in \mathbb {Z}\).

Proof

Let \(t_2,t_1 \in [0,t^*]\), \(t_2 > t_1\). We integrate (A.1) in time and estimate

$$\begin{aligned} |u(t_2,k)-u(t_1,k)|&\le \int _0^t |\Delta G(u_\delta )(s,k)| \ \mathrm {d}s \lesssim \int _{t_1}^{t_2} s^{\frac{\beta }{1-\beta }} \ \mathrm {d}s \end{aligned}$$
(A.30)
$$\begin{aligned}&\sim t_2^{\frac{1}{1-\beta }} - t_1^{\frac{1}{1-\beta }} \le (t_2-t_1)^\frac{1}{1-\beta }, \end{aligned}$$
(A.31)

where we used Lemma A.10 to estimate \(|\Delta G_\beta (u_\delta )|\). \(\square \)

With these preparations we can prove the first statement of Theorem A.3 for negative \(\beta \):

Proof of existence and positivity bound for \(\beta < 0\) Let \(u_{0,\delta }\) and \(u_\delta \) as above. By Corollary A.11 and the Arzela-Ascoli Theorem there exists a subsequence \(\delta \rightarrow 0\) such that \(u_\delta \rightarrow u\) uniformly on \([0,t^*]\). Moreover, by Lemma A.10 we have

$$\begin{aligned} u_\delta ^\beta (t,k) \le c^\beta t^{\frac{\beta }{1-\beta }}, \end{aligned}$$
(A.32)

which implies \(u_\delta ^\beta \rightarrow u^\beta \) in \(L^1([0,t^*])\). Using this we can pass to the limit in the integral equation

$$\begin{aligned} u_\delta (t,k) = u_{0,\delta }(k) - \int _0^t \Delta G(u_\delta )(s,k) \ \mathrm {d}s, \end{aligned}$$
(A.33)

showing that u is a solution to the backward equation with initial data \(u_0\) on \([0,t^*]\). Since the lower bound from Lemma A.10 also holds in the limit, we can extend the solution from \(t^*\) to arbitrary times via comparison principle, which also changes the lower bound to \(\sim 1 \wedge t^\frac{1}{1-\beta }\) for large times. \(\square \)

A.3 Harnack-Type Inequality for \(0 < \beta \le 1\)

In the previous part Lemma A.10 was crucial to prove existence of a solution. The result of the lemma, together with the positivity condition \(\mathcal {P}_{L,d}\) can be interpreted as a Harnack-type inequality, see Bonforte and Vazquez (2006). For \( 0< \beta < 1 \) a similar result holds, the equation however behaves differently and the indirect proof does not work here. We will pursue another approach and show the inequality directly with explicit constants, handling the case \(\beta = 1\) separately. The key observation is that a large particle next to a small particle will always induce growth on the small particle, despite the size of the other neighbour of the small particle. This decouples the equation in a sense and we only need to study the local problem:

Lemma A.12

(Local Problem) Let \(0< \beta < 1\) and \(T > 0\). Consider two functions \(F \in C^0([0,T];[0,\infty ))\) and \(u \in C^1([0,T];[0,\infty ))\) which satisfy

$$\begin{aligned} F(t)&\ge ct^{\frac{\beta }{1-\beta }}, \end{aligned}$$
(A.34)
$$\begin{aligned} \dot{u}(t)&\ge F(t) - 2u^\beta (t). \end{aligned}$$
(A.35)

on [0, T]. Then u satisfies

$$\begin{aligned} u(t) \ge \eta _1(c)t^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.36)

on the interval [0, T], where \(\eta _1\) is a positive strictly increasing function which depends only on \(\beta \).

Proof

We define the rescaled function

$$\begin{aligned} v(t) = t^{\frac{1}{\beta -1}}u(t), \end{aligned}$$
(A.37)

on the half-open interval (0, T]. Then it suffices to show that v is bounded from below by \(\eta _1(c)\). We calculate

$$\begin{aligned} t\dot{v}(t)&= t\left( t^{\frac{1}{\beta -1}}\dot{u}(t) + \frac{1}{\beta - 1}t^{\frac{1}{\beta -1}-1}u(t) \right) \end{aligned}$$
(A.38)
$$\begin{aligned}&\ge t\left( t^{\frac{1}{\beta -1}}F(t) -2t^{\frac{1}{\beta -1}}u^\beta (t) + \frac{1}{\beta - 1}t^{\frac{1}{\beta -1}-1}u(t) \right) \end{aligned}$$
(A.39)
$$\begin{aligned}&= t^{\frac{\beta }{\beta -1}}F(t) - 2v^\beta (t) - \frac{1}{1-\beta }v(t) \end{aligned}$$
(A.40)
$$\begin{aligned}&=: t^{\frac{\beta }{\beta -1}}F(t) - \theta (v(t)), \end{aligned}$$
(A.41)

in particular the assumption on F implies that

$$\begin{aligned} t\dot{v}(t) \ge c - \theta (v(t)). \end{aligned}$$
(A.42)

Since the function \(\theta \) is strictly increasing on \([0,\infty )\) we can define the inverse function \(\eta _1 = \theta ^{-1}\), which is also strictly increasing. We claim that \(v \ge \eta _1(c)\) on (0, T]. If this is not true, there is \(\varepsilon > 0\) and \(t^* \in (0,T]\) such that \(v(t^*) \le \eta _1(c) - \varepsilon \). In particular we have

$$\begin{aligned} c - \theta (v(t^*)) \ge \tilde{\varepsilon } > 0, \end{aligned}$$
(A.43)

for some \(\tilde{\varepsilon } > 0\). But then the differential inequality (A.42) implies that \(v(t) \le \eta _1(c) - \varepsilon \), and hence \(c - \theta (v(t)) \ge \tilde{\varepsilon }\) for all \( t \in (0,t^*]\). Dividing by t and integrating (A.42) in time gives

$$\begin{aligned} v(t^*) - v(t) \ge \varepsilon \log \left( \frac{t^*}{t} \right) , \end{aligned}$$
(A.44)

for all \(0 < t \le t^*\). Sending t to zero then gives a contradiction. \(\square \)

The above lemma enables us to prove a Harnack-type inequality:

Lemma A.13

(Harnack-type inequality) Let u be a solution to Eq. (A.1) with \(0< \beta < 1\) and initial data \(0 \le u_0 \le 1\). Then we have

$$\begin{aligned} u(t,k) \ge \eta (|k-l|)(t-s)^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.45)

for all \(k,l \in \mathbb {Z}\) and \(0 \le t -s \le t^*(u(s,l))\). The function \(\eta \) is strictly positive and the function \(t^*\) is nonnegative, strictly increasing with \(t^*(u) = 0\) iff \(u=0\). Furthermore, both functions depend only on \(\beta \).

Proof

Due to translation invariance in space and time it suffices to consider the case \(s=0\) and \(l=0\). We will make an iterative argument, using Lemma A.12 in each step. First we note that due to \(u_0 \le 1\) and the comparison principle we have the Lipschitz estimate

$$\begin{aligned} |\dot{u}| \le 4, \end{aligned}$$
(A.46)

in particular

$$\begin{aligned} u(t,0) \ge u_0(0) - 4t. \end{aligned}$$
(A.47)

The case \(u_0(0) = 0\) is trivial. If \(u_0(0) > 0\), the Lipschitz estimate implies

$$\begin{aligned} u(t,0) \ge t^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.48)

whenever \(u_0(0) - 4t - t^{\frac{1}{1-\beta }} > 0\). If we set \(f(t) = 4t + t^{\frac{1}{1-\beta }}\) then \(t^*\) is defined as the inverse of f. Thus the above lower bound holds for \(0 \le t \le t^*(u_0(0))\). For u(t, 1) we have

$$\begin{aligned} \dot{u}(t,1)&= u^\beta (t,0) - 2u^\beta (t,1) + u^\beta (t,2) \end{aligned}$$
(A.49)
$$\begin{aligned}&\ge u^\beta (t,0) - 2u^\beta (t,1). \end{aligned}$$
(A.50)

This means that \(u^\beta (t,0)\) and u(t, 1) satisfy the assumptions of Lemma A.12 with \(T = t^*(u_0(0))\) and \(c = 1\). Thus we have

$$\begin{aligned} u(t,1) \ge \eta _1(1)t^{\frac{1}{1-\beta }}, \end{aligned}$$
(A.51)

on \([0,t^*(u_0(0))]\). Now we can successively apply the same argument to the pairs of functions \((u^\beta (t,1)\),\(u(t,2)),\ldots ,(u^\beta (t,k-1)\),u(tk)), for \(k \in \mathbb {N}\). The argument for \(-k\) is the same. Then the desired inequality follows with the function \(\eta = \eta (r)\) (\(r \in \mathbb {N}\)) defined as

$$\begin{aligned} \eta (r)&= \eta _1^{(r)}(1), \end{aligned}$$
(A.52)
$$\begin{aligned} \eta (0)&= 1, \end{aligned}$$
(A.53)

where \(\eta _1^{(r)}\) means that \(\eta \) is r times composed with itself. \(\square \)

Lemma A.14

Let u be a solution to the constant coefficient linear equation (A.1) with \(\beta = 1\). Then for every \(k \in \mathbb {Z}\) and \(N \in \mathbb {N}\) we have

$$\begin{aligned} u(t,k) \ge M(u_0,k,N) \exp (-2t)I_N(2t), \end{aligned}$$
(A.54)

where \(I_N(t)\) denotes the N-th modified Bessel function of the first kind and

$$\begin{aligned} M(u_0,k,N) = \sum _{l=-N}^N u_0(k-l) \end{aligned}$$
(A.55)

denotes the local initial mass.

Proof

In the linear constant coefficient case we can give an explicit formula by Fourier-analysis: We write

$$\begin{aligned} \hat{u}(t,\theta ) = \sum _{k=-\infty }^{k=+\infty } u(t,k)\exp (-ik\theta ), \end{aligned}$$
(A.56)

taking the time derivative on both sides and using the equation then yields

$$\begin{aligned} \partial _t \hat{u}(t,\theta )&= \exp (-i\theta )\sum _{k=-\infty }^{k=+\infty } u(t,k)\exp (-ik\theta ) \end{aligned}$$
(A.57)
$$\begin{aligned}&\quad -\, 2\sum _{k=-\infty }^{k=+\infty } u(t,k)\exp (-ik\theta ) \end{aligned}$$
(A.58)
$$\begin{aligned}&\quad +\, \exp (i\theta )\sum _{k=-\infty }^{k=+\infty } u(t,k)\exp (-ik\theta ) \end{aligned}$$
(A.59)
$$\begin{aligned}&= 2(\cos (\theta ) - 1)\hat{u}(t,\theta ). \end{aligned}$$
(A.60)

We solve this ODE in t with initial data \(\hat{f}\) to obtain

$$\begin{aligned} \hat{u}(t,\theta ) = \hat{f}(\theta )\exp (2t(\cos (\theta ) - 1)), \end{aligned}$$
(A.61)

which gives the discrete heat kernel

$$\begin{aligned} \phi (t,k)&= \exp (-2t) \frac{1}{2\pi } \int _{-\pi }^\pi \exp (2t\cos (\theta )-ik\theta ) \ \mathrm {d}\theta \end{aligned}$$
(A.62)
$$\begin{aligned}&= \exp (-2t) \frac{1}{\pi } \int _{0}^\pi \exp (2t\cos (\theta ))\cos (k\theta ) \ \mathrm {d}\theta \end{aligned}$$
(A.63)
$$\begin{aligned}&= \exp (-2t)I_k(2t), \end{aligned}$$
(A.64)

where \(I_k\) is the kth modified Bessel function of the first kind. Then the desired inequality follows directly by the standard representation

$$\begin{aligned} u(t,k) = \sum _{l \in Z}u_0(k-l)\phi (t,l), \end{aligned}$$
(A.65)

the fact that \(\phi \) is decreasing in the second argument and the obvious estimate. \(\square \)

We summarise the findings of this section and prove the remaining statements of Theorem A.3:

Proof of positivity estimate for \(0 < \beta \le 1\) First we consider the case \(\beta \ne 1\). It suffices to consider the case \(u_0 \le 1\) by scaling (this means that for general data the constants get an additional dependence on \(||u_0||_\infty \)). Because \(u_0 \in \mathcal {P}_{L,d}\), for every \(k \in \mathbb {Z}\) there exists \(k'\) with \(u_0(k') \ge d\) and \(|k-k'| \le L\). Then Lemma A.13 with \(s=0\) and \(l = k'\) yields

$$\begin{aligned} u(t,k) \ge \min _{j=1,\ldots ,L}\eta (j) t^\frac{1}{1-\beta }, \end{aligned}$$
(A.66)

for \(0 \le t \le t^*(d)\) because \(t^*\) is monotone. For \(\beta = 1\) we note that \(u_0 \in \mathcal {P}_{L,d}\) implies that \(M(u,k,L) \ge 2d\), since there are at least two terms in the sum that are greater than or equal to d by definition of \(\mathcal {P}_{L,d}\). Then the statement follows directly from Lemma A.14. \(\square \)

A.4 Nash–Aronson Estimates and Hölder Continuity

For \(u = u(k) \in \ell ^\infty (\mathbb {Z})\) we define the forward and backward difference operators

$$\begin{aligned} \partial ^+ u(k)&= u(k+1) - u(k) \end{aligned}$$
(A.67)
$$\begin{aligned} \partial ^- u(k)&= u(k) - u(k-1). \end{aligned}$$
(A.68)

For \(a = a(t,k)\) with \(0 < c_1 \le a \le c_2\) and \(a(.,k) \in C^0([0,\infty ))\) we consider the discrete analogue to a parabolic evolution equation in divergence form:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u = \partial ^-(a \partial ^+ u) =: \mathcal {L}(t)u \\ u(0,.) = u_0. \end{array}\right. } \end{aligned}$$
(A.69)

We denote by \(\phi (t,k,s,l)\) the fundamental solution to (A.69), in other words, \(\phi (.,.,s,l)\) is the solution to the above equation starting at time s with initial data \(\phi _0(k) = \delta _{kl}\). Since \(\mathcal {L}(t)\) is a bounded operator from \(\ell ^2(\mathbb {Z})\) to \(\ell ^2(\mathbb {Z})\), \(\phi \) can be written as

$$\begin{aligned} \phi (t,k,s,l) = \left\langle \exp \left( \int _s^t \mathcal {L}(r) \ \mathrm {d}r \right) \delta _l, \delta _k \right\rangle , \end{aligned}$$
(A.70)

where \(\delta _k\) are the canonical basis vectors in \(\ell ^2(\mathbb {Z})\). The general solution starting at time \(t = s\) to (4.22) is then given by

$$\begin{aligned} u(t,k) = \sum _{l \in \mathbb {Z}^d}\phi (t,k,s,l)u_0(l). \end{aligned}$$
(A.71)

We also define the reduced fundamental solution

$$\begin{aligned} \psi (t,k)&= \phi (t,k,0,0) = \phi (t,0,0,k), \end{aligned}$$
(A.72)

and the corresponding “macroscopic” rescaled function

$$\begin{aligned}&U:\mathbb {R}\rightarrow [0,+\infty ) \end{aligned}$$
(A.73)
$$\begin{aligned}&U(t,\xi ) = t^\frac{1}{2}\psi (t,\lfloor t^\frac{1}{2} \xi \rfloor ). \end{aligned}$$
(A.74)

We have the following Nash-Aronson estimates on the fundamental solution:

Theorem A.15

There exist constants \(t_0 > 0\), \(C > 0\) and \(\alpha > 0\), depending only on the bounds on a, such that the following statements hold:

  • Aronson estimate:

    $$\begin{aligned} \psi (t,k)&\le \frac{C}{1 \vee t^\frac{1}{2}}\exp \left( -\frac{|k|}{1 \vee t^\frac{1}{2}} \right) , \end{aligned}$$
    (A.75)

    for every \(k \in \mathbb {Z}\) and \(t \ge 0\).

  • Nash continuity estimate:

    $$\begin{aligned} |\psi (t,k) - \psi (t,l)|&\le \frac{C}{t^\frac{1}{2}}\left( \frac{|k-l|}{t^\frac{1}{2}} \right) ^\alpha , \end{aligned}$$
    (A.76)

    for every \(k,l \in \mathbb {Z}\) and \(t \ge 0\).

Proof

Here we cite the results from Appendix B of Giacomin et al. (2001). Inequality (A.75) is precisely the statement of Proposition B.3. For the second inequality (A.76) we first note that (A.75) implies \(|\psi | \lesssim t^{-\frac{1}{2}}\). Then the desired estimate at a time \(t^*\) follows from Proposition B.6 applied at \(t = s = t^*/2\) with \(f = \psi (t^*/2,.)\) and the semigroup property. \(\square \)

These estimates have important consequences for the function U. Inequality (A.75) implies that

$$\begin{aligned} U(t,\xi ) \le \Phi (\xi ), \end{aligned}$$
(A.77)

for some integrable function \(\Phi \). In particular the function family U(t, .) are tight probability measures. On the other hand, the estimate (A.76) implies that the function U(t, .), which is a step-function by definition, becomes Hölder continuous in the following sense:

Definition A.16

(Approximate Hölder Continuity) Let \(\{f_n \} \subset L^\infty (\mathbb {R})\) be a sequence of functions. Then \(\{f_n \}\) is said to be approximately Hölder continuous with exponent \(\alpha \in (0,1]\) if for every \(\varepsilon > 0\) there exists \(n = n(\varepsilon )\) such that \(|x-y| \ge \varepsilon \) implies

$$\begin{aligned} |f_n(x) - f_n(y)| \le C |x-y|^\alpha , \end{aligned}$$
(A.78)

for \(n \ge n(\varepsilon )\) and a universal positive constant C.

The important observation is that Hölder continuity on the discrete microscopic level implies approximate Hölder continuity on the macroscopic scale:

Lemma A.17

The function U(t, .) is approximately Hölder continuous as \(t \rightarrow \infty \). Furthermore, the constants \(C, \alpha \) and \(t = t(\varepsilon )\) only depend on the bounds of the coefficient a in (A.69).

Proof

By the estimate (A.76) from Theorem A.15 we calculate

$$\begin{aligned} |U(t,\xi )-U(t,\eta )| \lesssim \left( \frac{|\lfloor t^\frac{1}{2} \xi \rfloor - \lfloor t^\frac{1}{2} \eta \rfloor |}{t^\frac{1}{2}} \right) ^\alpha . \end{aligned}$$
(A.79)

Since

$$\begin{aligned} \frac{|\lfloor t^\frac{1}{2} \xi \rfloor - \lfloor t^\frac{1}{2} \eta \rfloor |}{t^\frac{1}{2}} = |\xi - \eta | + \mathcal {O}(t^{-\frac{1}{2}}), \end{aligned}$$
(A.80)

we get the desired estimate for \(|\xi - \eta | \gtrsim t^{-\frac{1}{2}} \). \(\square \)

The next result is of crucial importance for the main result of the paper. Denote by \(\mathcal {T}_\delta (\mathbb {R})\) the set of step-functions with step-width at least \(\delta \). Then we have:

Lemma A.18

Let \((f_n) \subset L^1(\mathbb {R})\) be tight and approximately Hölder continuous. Then for every \(\varepsilon > 0\) there exists \(n_0\) and \(\delta > 0\), such that for every \(f_n\) with \(n \ge n_0\) there exists \(\chi \in \mathcal {T}_\delta (\mathbb {R})\) with

$$\begin{aligned} ||f_n - \chi ||_{L^1(\mathbb {R})} \le \varepsilon . \end{aligned}$$
(A.81)

Proof

Let \(\varepsilon > 0\). Because \((f_n)\) is tight in \(L^1\) there exists \(R > 0\) such that

$$\begin{aligned} \int _{|x| \ge R} |f_n(x)| \ \mathrm {d}x \le \varepsilon , \end{aligned}$$
(A.82)

hence it suffices to approximate \((f_n)\) in \(L^\infty \). By approximate Hölder continuity there exists \(n_0\) such that

$$\begin{aligned} |f_n(x) - f_n(y)| \le C |x-y|^\alpha , \end{aligned}$$
(A.83)

for \(n \ge n_0\) and \(|x-y| \ge R^{-\frac{1}{\alpha }}\varepsilon \). This means that the piecewise-constant interpolation \(\chi \) of \(f_n\) with step-width \(R^{-\frac{1}{\alpha }}\varepsilon \) approximates \(f_n\) uniformly up to an error of \(R^{-1}\varepsilon ^\alpha \), hence

$$\begin{aligned} ||f_n - \chi ||_{L^1} \lesssim \varepsilon + \varepsilon ^\alpha . \end{aligned}$$
(A.84)

\(\square \)

Combining the last two lemmas we obtain the following corollary, which is used in the proof of the main result:

Corollary A.19

For every \(\varepsilon > 0\) there exists \(T > 0\) and \(\delta > 0\), such that for every U(t, .) with \(t \ge T\) there exists \(\chi \in \mathcal {T}_\delta (\mathbb {R})\) with

$$\begin{aligned} ||U(t,.) - \chi ||_{L^1(\mathbb {R})} \le \varepsilon . \end{aligned}$$
(A.85)

Furthermore, T and \(\delta \) only depend on \(\varepsilon \) and the bounds on a.

Proof

Approximate Hölder continuity was already established, while tightness in \(L^1\) follows from the estimate (A.75) of Theorem A.15. The dependence of the constants is easily checked revisiting the proofs of the previous two lemmas. \(\square \)

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Eichenberg, C. Special Solutions to a Nonlinear Coarsening Model with Local Interactions. J Nonlinear Sci 29, 1343–1378 (2019). https://doi.org/10.1007/s00332-018-9519-1

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Keywords

  • Coarsening
  • Infinite particle system
  • Backward fast diffusion
  • Discrete parabolic regularity

Mathematics Subject Classification

  • 70F45
  • 35K55
  • 37L60