A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term −Δu_t is added, and it is well known that this equation is well posed in the space of W^1,2 functions. In this paper, the viscosity term is changed to , and it is proved that if initial data is slightly smooth (but belonging to W^2,2 is sufficient), then a weak solution exists uniquely in the space of BV functions.

1. Introduction

Let Ω be a bounded domain in Rⁿ with the Lipschitz continuous boundary ∂Ω. In [1] and in the author’s previous works [2–4], the following:

()

is investigated, which is in these works referred to as the equation of motion of vibrating membrane. Up to now, neither existence nor uniqueness of a solution to (1) is obtained. In [1–3], we only have that a sequence of approximate solutions to (1) converges to a function u in an appropriate function space, and that if u satisfies the energy conservation law, it is a weak solution to (1). In [1], approximate solutions are constructed by the Ritz-Galerkin method and in [2, 3] by Rothe’s method. In [2], the boundary condition is not essentially discussed, and the observation is added in [3]. In these works, the limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet. Instead, in [4], linear approximation for (1) is established. On the other hand, the equation with the strong viscosity term −Δu_t is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if

and u_t(0) ∈ L²(Ω), there exists a unique solution

for each T > 0. Namely, the equation with strong viscosity term is well posed in W^1,2(Ω), and since W^1,2 is a smaller class than the space of BV functions, this suggests that the influence of the term −Δu_t is too strong.

In this paper, replacing the strong viscosity term −Δu_t with

, we investigate it in the space of BV functions. Namely, our problem of this paper is as follows:

()

with initial and boundary conditions

()

We should note that the term “viscosity” probably means implying regularity. However, in this paper, we only investigate existence and uniqueness of (2)–(4), regularity is not investigated. This is the reason that in the title there is a quotation mark.

A function u is said to be a function of bounded variation or a BV function in Ω if the distributional derivative Du is an Rⁿ valued finite Radon measure in Ω. The vector space of all functions of bounded variation in Ω is denoted by BV(Ω). It is a Banach space equipped with the norm

(see, e.g., [6–8]). We should note that, for u ∈ BV(Ω), the operator

is multivalued. It is usually defined by the use of the subdifferential of the area functional. Namely, for a function u ∈ BV(Ω)∩L²(Ω), we regard

()

where

()

Here, readers should note that J(u) is not

. We are imposing (4), and in the analysis in the space of BV functions, the most appropriate weak formulation of (4) is to replace

with

(cf. [3], see, also [4, Appendix C]).

Now, we present our definition of a weak solution to (2)–(4).

Definition 1. A function u is a weak solution to (2)–(4) in [0, T) × Ω if u satisfies that

(i)
u ∈ L^∞((0, T); BV(Ω)∩L²(Ω)), u_t ∈ L²((0, T) × Ω),
(ii)
s-lim _t↘0u(t, x) = u₀(x) in L²(Ω),
(iii)
there exist f₀ ∈ ∂J(u₀) and such that for ℒ¹-a.e. t (ℒ¹ denotes the one-dimensional Lebesgue measure), f₁ ∈ ∂J(u) and, for any ,
()

If a function u ∈ L^∞((0, ∞); BV(Ω)∩L²(Ω)) is a weak solution to (2)–(4) in [0, T) × Ω for each T > 0, then we say that u is a weak solution to (2)–(4) in [0, ∞) × Ω.

Our main theorem is as follows.

Theorem 2. Suppose that u₀ ∈ BV(Ω)∩L²(Ω) and v₀ ∈ L²(Ω). We further suppose that J(u₀)∩L²(Ω) ≠ ∅. Then, there exists a unique weak solution to (2)–(4) in [0, ∞) × Ω.

Remark 3. If J(u₀)∩L²(Ω) ≠ ∅, then the element is unique. Indeed, for each , J(u₀ + εϕ) is differentiable at ε = 0 and (d/dε)J(u₀ + εϕ)|_ε=0 = (f₀, ϕ) for each f₀ ∈ J(u₀)∩L²(Ω). Since f₀ ∈ L²(Ω) and ϕ is arbitrary, f₀ is uniquely determined.

2. Reduction of the Problem

In order to solve (2)–(4), we give a formal observation. Let us put

()

then (2) becomes f_t = −f − g, which can be regarded as an ordinary differential equation to f. By the variation-of-constants formula, we obtain that

. Noting that g = u_tt, we have

()

where

. Hence, formally, (2) is reduced to

()

Definition of a weak solution to this equation is as follows.

Definition 4. Let f₀ ∈ ∂J(u₀), u₀ ∈ BV(Ω)∩L²(Ω), and v₀ ∈ L²(Ω). A function u is a weak solution to (10) with (3) and (4) in [0, T) × Ω if u satisfies that

(i)
u ∈ L^∞((0, ∞); BV(Ω)∩L²(Ω)), u_t ∈ L²((0, T) × Ω),
(ii)
s-lim _t↘0u(t, x) = u₀(x) in L²(Ω),
(iii)
for any ϕ ∈ L²(Ω)∩BV(Ω) and for ℒ¹-a.e. t,
()

Similar to the case of (2), we say that u is a weak solution to (10) with (3) and (4) in [0, ∞) × Ω if a function u ∈ L^∞((0, ∞); BV(Ω)∩L²(Ω)) is a weak solution to (10) with (3) and (4) in [0, T) × Ω for each T > 0.

The previous observation is just formal. In the following proposition, we show it rigorously.

Proposition 5. Definitions 1 and 4 are equivalent.

Proof. It is sufficient to show that, for each T > 0, a function u is a weak solution to (2)–(4) in [0, T) × Ω if and only if it is a weak solution to (10) with (3) and (4) in [0, T) × Ω.

Suppose that u is a weak solution to (10) with (3) and (4) in [0, T) × Ω. Conditions (i) and (ii) of Definition 1 are the same as those of Definition 4. Thus, we only have to show (iii) of Definition 1. Let

()

Then, by (iii) of Definition 4, we have that f₁ ∈ ∂J(u) for ℒ¹-a.e. t. Thus, by a direct calculation, we have that u satisfies (iii) of Definition 1.

Next, we suppose that u is a weak solution to (2)–(4) in [0, T) × Ω. For each ϕ ∈ L^∞((0, T); L²(Ω)∩BV(Ω)) and each , we put

()

Then

, and since ψ_t(t) = ψ(t) − ρ(t)ϕ(t), we have the following by (iii) of Definition 1:

()

By integration by parts, we have

()

Furthermore, we have the following by Fubini’s theorem:

()

Finally, noting that

, we have the following by (14), (15), and (16):

()

Since ϕ and ρ are arbitrary, we have, for ℒ¹-a.e. t,

()

which means that u satisfies Definition 4, (iii).

Now, Theorem 2 is reduced to the following.

Theorem 6. Suppose that u₀ ∈ BV(Ω)∩L²(Ω) and v₀ ∈ L²(Ω). We further suppose that J(u₀)∩L²(Ω) ≠ ∅ and let f₀ ∈ J(u₀)∩L²(Ω). Then, there exists a unique weak solution to (10), (3), and (4) in [0, ∞) × Ω.

Our strategy of proving Theorem 6 is the contracting mapping theorem. For this purpose, given that

, we solve

()

and show that the map

is a contraction. A weak solution to (19) with (3) and (4) is defined as follows.

Definition 7. Let f₀ ∈ ∂J(u₀). A function u is a weak solution to (19) with (3) and (4) in [0, T) × Ω if u satisfies that

(i)
u ∈ L^∞((0, T); BV(Ω)∩L²(Ω)), u_t ∈ L²((0, T) × Ω),
(ii)
s-lim _t↘0u(t, x) = u₀(x) in L²(Ω),
(iii)
for any ϕ ∈ L²(Ω)∩BV(Ω) and for ℒ¹-a.e. t,
()

The proof of Theorem 6 consists of two parts. The first part is solving (19), and the second part is to show that the map is a contraction.

3. Existence and Uniqueness of a Solution to (19)

Let u₀ and v₀ be as in Theorem 6. In this section, we show that there exists a unique solution to (19) with (3) and (4) in [0, T) × Ω for each T > 0.

Uniqueness is easy. Suppose that u and v are solutions to (19) with (3) and (4) in [0, T) × Ω, and inserting ϕ = v − u to (iii) of Definition 7, integrating it from 0 to T, obtaining another inequality by replacing u and v, and adding these two inequalities, we have

()

Since u(0) = v(0) = u₀, we have the uniqueness of a solution to (19).

It is sufficient to show the existence in [0, T) × Ω for ℒ¹-a.e. T. Approximate solutions are constructed by Rothe’s time semidiscretization method. In Rothe’s method, we should solve elliptic equations with respect to space variables. Here, we solve them by a direct variational method (namely, this is the method of discrete Morse semiflow, cf. [9] and references cited therein).

Suppose that u₀ ∈ BV(Ω)∩L²(Ω) with J(u₀)∩L²(Ω) ≠ ∅ and v₀ ∈ L²(Ω), and let f₀ ∈ J(u₀)∩L²(Ω). For a positive number h, we construct a sequence

in the following way. For l = 0, we let u₀ be as in (3), and for l ≥ 1, it is defined as a minimizer of the following functional:

()

in the class L²(Ω)∩BV(Ω), where

()

Since

()

F_l is bounded from below, and hence, the existence of a minimizer of ℱ_l follows.

Lemma 8 (energy inequality). consider the following:

()

Proof. Since u_l is a minimizer of ℱ_l, we have

()

Hence, for each l,

()

Thus, by induction on l, we have the conclusion.

Next, we define approximate solutions u^h(t, x) and

for (t, x)∈(−h, ∞) × Ω as follows: for (l − 1)h < t ≤ lh,

()

Then Lemma 8 shows for each T > 0

()

Now, we estimate the second term of the left hand side of (29). Then,

()

where C(T) = 2⁻¹(T + 2⁻¹e^−2T), and thus, it is easy to see that

()

By (29), we have, for each ε > 0 and for each T > 0,

()

where

()

Proposition 9. It holds that

(1)
is uniformly bounded with respect to h;
(2)
for any T > 0, is uniformly bounded with respect to h;
(3)
for any T > 0, is uniformly bounded with respect to h;
Then there exist a sequence {h_j} with h_j → 0 as j → ∞ and a function u such that
(4)
converges to u_t as j → ∞ weakly in L²((0, ∞) × Ω);
(5)
for any T > 0, converges to u as j → ∞ weakly star in L^∞((0, T); L²(Ω));
(6)
for any T > 0, converges to u as j → ∞ strongly in L^∞((0, T); L^p(Ω)) for each 1 ≤ p < 1^* = n/(n − 1);
(7)
for any T > 0, converges to u as j → ∞ strongly in L^∞((0, T); L^p(Ω)) for each 1 ≤ p < 1^*;
(8)
u ∈ L^∞((0, ∞); BV(Ω));
(9)
for ℒ¹-a.e. t ∈ (0, ∞), converges to Du(t, ·) as j → ∞ in the sense of distributions;
(10)
s- lim _t↘0u(t) = u₀ in L²(Ω).

Proof. Assertion (1) immediately follows from (32). Since we have

()

for each t, t^′ ≥ 0, Assertion (1) implies that, for each T > 0,

is uniformly bounded with respect to h. Given that t > 0, we let l be an integer such that (l − 1)h < t ≤ lh. Then,

()

By (32),

()

Thus, we have

()

for each l, where

. Hence,

()

Now, we have that

is uniformly bounded with respect to h since

()

Since C₀ is increasing with respect to T, Assertion (2) follows from (32). Since J is convex, we have

()

and Assertion (3) also holds.

Assertion (4) is a direct consequence of Assertion (1). Assertion (5) follows from Assertion (3). Furthermore, (34) and Assertion (1) imply that the function t ↦ u^h(t, ·) ∈ L²(Ω) is equicontinuous with respect to h. By Sobolev’s theorem BV(Ω) ⊂ L^p(Ω) compactly for each 1 ≤ p < 1^*. This means that, for any T > 0, {u^h(t, ·)} is contained in a sequentially compact subset of L^p(Ω) which is independent of h and t ∈ [0, T]. Thus, by the Ascoli-Arzela theorem, we obtain Assertion (6).

Now, we have, for 1 ≤ p < 1^*,

()

the right hand side of which converges to 0 as h → 0 by (38) and Assertion (6). Now, we have Assertion (7). Assertions (2) and (7) imply Assertions (8) and (9).

Letting t^′ = 0 in (34), we have

()

Thus, by Assertion (1) the left hand side is uniformly bounded with respect to h and, hence, passing to a subsequence if necessary, {u^h(t) − u₀} _h>0 converges weakly in L²(Ω), and by Assertion (6), the weak limit is u(t) − u₀. Then, by the lower semicontinuity of L² norm, we have

()

which implies Assertion (10).

Now, our purpose is to show that u is a weak solution to (19). Proposition 9 implies that u satisfies (i) and (ii) of Definition 7.

Since u_l is a minimizer of ℱ_l(v), we have

()

Let us write, for (l − 1)h ≤ t < lh,

. Then, for each j and for ℒ¹-a.e. t ∈ (0, ∞),

()

namely, for each v ∈ L²(Ω),

()

By Proposition 9 (7) we have that, for ℒ¹-a.e. t ∈ (0, ∞),

()

strongly in L¹(Ω) as j → ∞. Let T be a number t such that (46) and (47) hold. We insert an arbitrary function v ∈ L²((0, T) × Ω) in (46). Integrating it from 0 to T, we have the following by Proposition 9 (4), (7), Fatou’s lemma, and the lower semicontinuity:

()

For a while, we write h_j = h for simplicity. First, we note the following identity:

()

Let L be the integer such that (L − 1)h < T ≤ Lh. By (49), we have

()

By (42) and Proposition 9 (1),

is uniformly bounded with respect to t ∈ [0, T] and h. On the other hand, since (L − 1)h < T ≤ Lh,

()

Hence, we have by (37)

()

and thus by (37) again

()

as h → 0. By Proposition 9 (1),

()

as h → 0. Since

, u(T) ∈ L²(Ω), (47) implies that

in L²(Ω). In particular, we have

. Then, since

, we have

()

Summing up, we have

()

It is not difficult to show that G^h → G strongly in L²((0, T) × Ω). Hence, by (48), we finally have

()

for each v ∈ L²((0, T) × Ω). By the convexity of J, for each function ψ ∈ L²(0, T) with 0 ≤ ψ ≤ 1, we have ψJ(v) − ψJ(u) ≥ J(u + ψ(v − u)) − J(u). Thus, (57) implies that

()

It is easy to extend this inequality to all nonnegative functions ψ ∈ L²(0, T). Hence, (iii) of Definition 7 holds for ℒ¹-a.e. t ∈ (0, ∞).

4. Proof That Is a Contraction

Let

be functions in L²((0, T) × Ω). In this section, we write

()

Then,

()

Let u, v be a solution to (19) with (3) and (4) for

, respectively. By (iii) of Definition 7,

()

Summing these, we have

()

Integrating from 0 to t, we have the following by (60) and by the fact that u(0) = v(0) = u₀:

()

We further integrate this from 0 to T and write it I + II + III. Then,

()

Here,

()

Thus,

()

These two terms are estimated as follows:

()

Summing up, we have

()

Hence, when 0 < T < 1/2, putting

()

we have

()

As T → 0, K(T) converges to 0. Thus, if T is sufficiently small, 0 < K(T) < 1. This means that the map from

to u is a contraction in L²((0, T) × Ω). Hence, there is a fixed point

and it is a solution to (10) with (3) and (4) in [0, T) × Ω.

End of the proof of Theorem 6.

Uniqueness of a Local Solution. Let

be solutions to (10) with (3) and (4). Then, in the same calculus as before, we obtain

()

This implies the uniqueness.

Existence of a Time Global Solution. Suppose that

is a solution to (10) with (3) and (4) in [0, s]. First, we remark that

()

and the right hand side belongs to L²(Ω). Hence, we are able to solve (10) with (3) and (4) from t = s. By the change of variable

, (10) becomes

()

where

()

Hence, solving (10) with (4) and initial condition

, we obtain a function u in [0, T] × Ω which solves (10). For t ∈ [s, s + T], put

. Then,

is a solution to (10) with (3) and (4) in [0, s + T] × Ω. Repeating this process, we obtain a time global solution. Uniqueness of the local solution implies uniqueness of the global solution.

5. Uniform Estimates

Let T be the small number presented in the previous section, and let u be a solution to (10) with (3) and (4) in [0, T) × Ω. As we see in the previous section, it is obtained as a fixed point of the map

in L²((0, T) × Ω), where u is a solution to (19) with (3) and (4). The fixed point is obtained as in the following way. Let u⁽⁰⁾ be an arbitrary element of L²((0, T) × Ω) and put τⁿ(u⁽⁰⁾) = u⁽ⁿ⁾. There is a constant σ which is determined by T such that 0 < σ < 1 and

. Thus, we have

()

and hence,

()

namely, {u⁽ⁿ⁾} is a Cauchy sequence in L²((0, T) × Ω) and it converges to a function u. Since u⁽ⁿ⁾ = τ(u⁽ⁿ⁻¹⁾), we have that u = τ(u) by letting n → ∞. Thus, u is the fixed point of τ.

Letting m = 0 in (78), we have

()

By the lower semicontinuity, an energy inequality for a solution to (19) with (3) and (4) is obtained by letting h → 0 in (32):

()

Since C₀ is increasing with respect to T, we have ess. sup _0<t≤TJ(u(t, ·)) ≤ C₀(T). Hence,

()

Recall that C₀ is presented as in (33). By the proof of Proposition 9 (5), we have

()

Letting n → ∞ in (79), we finally have

()

Now, a solution u to (10) is a solution to (19) for

, the fixed point of τ. Hence, by (81), we have

()

By (83) and (84), there exists a constant C₃ such that

()

Now, by (83) and (85), we obtain uniform estimates for

for the small T.

Solving (10) from t = s (0 < s < T), we have the following by (83):

()

where U_s is as in (76). By (81) and Chebyshev’s inequality, for sufficiently large R,

. Let δ be an arbitrary small positive number and put

. Then, there exists an s ∈ [T − δ, T) = [T − (1/R²)(4C₀/(1 − ε), T)) such that

. Hereby, we have by (83) and (85) that, for such an s, there exists a constant C₄ such that

()

Repeating this process, we have uniform estimates for

for each

. However, their upper bounds depend on

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Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

Abstract

1. Introduction

2. Reduction of the Problem

3. Existence and Uniqueness of a Solution to (19)

4. Proof That Is a Contraction

5. Uniform Estimates

References

References

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Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

Abstract

1. Introduction

2. Reduction of the Problem

3. Existence and Uniqueness of a Solution to (19)

4. Proof That Is a Contraction

5. Uniform Estimates

References

References

Related

Information