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Solvability of one non-Newtonian fluid dynamics model with memory V.G. Zvyagin*, V.P. Orlov Laboratory of mathematical fluid dynamics of Research Institute of Mathematics, Voronezh State University, Universitetskaya pl. 1, 394018, Voronezh, Russia

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Article history: Received 18 May 2017 Accepted 28 February 2018 Communicated by Enzo Mitidieri MSC: primary 35A01 secondary 35Q35 45K05 76D03 Keywords: Viscoelastic fluid Initial–boundary value problem Topological approximation method Topological degree Measure of noncompactness Regular Lagrangian flow

In the present paper we establish the existence of weak solutions to the initial– boundary value problem for one viscoelastic model of Oldroyd’s type fluid with memory along trajectories of the velocity field. Previously such problem has been studied for corresponding regularized models. The reason of the regularization was the lack of results on the solvability of the Cauchy problem with not sufficiently smooth velocity field. However, recent results about regular Lagrangian flows (generalization of classical solutions of a Cauchy problem) allow to establish the existence theorem for the original problem. We use topological approximation method which involves the approximation of the original problem by regularized operator equations with consequent application of topological degree theory for its solvability. This allows to establish the existence of weak solutions of considered problem on the base of a priori estimates and passing to the limit. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction We consider the motion of a fluid that occupies a bounded domain Ω ⊂ Rn , n = 2, 3, with locally Lipschitz boundary ∂Ω on a time interval [0, T ], T > 0. The motion equation in the Cauchy form (see [11], Ch. II, Sec. 4–6) is ρ(∂v(t, x)/∂t +

n ∑

vi (t, x) ∂v(t, x)/∂xi ) =

i=1

− ∇ p(t, x) + Div σ(t, x) + ρf (t, x), (t, x) ∈ Q = [0, T ] × Ω ,

(1.1)

where v = (v1 (t, x), . . . , vn (t, x)) is the velocity at a point x ∈ Ω at time t; ρ is the fluid density; p = p(t, x) is the pressure; σ = σ(t, x) = {σij (t, x)}ni,j=1 is the deviator of the stress tensor; f = f (t, x) is the density

*

Corresponding author. E-mail addresses: zvg [email protected] (V.G. Zvyagin), orlov [email protected] (V.P. Orlov).

https://doi.org/10.1016/j.na.2018.02.012 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

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of external forces; Div σ is a vector function whose coordinates are divergences of the rows of the matrix σ. Below ρ supposed to be equal to 1 for simplicity. Eq. (1.1) is completed by a constitutive law (rheological relation) defining the type of a fluid (see [16]). The constitutive law σ = 2νE(v) defines the Newtonian fluid. Here E(v) ={Eij }ni,j=1 , Eij = 12 (∂vi /∂xj + ∂vj /∂xi ) is the strain rate tensor. The well known Navier–Stokes system corresponds to this law. However, there is a wide range of viscous incompressible non-Newtonian fluids (see e.g. [1,4,5,12,15,22,23,25,26]), among them are Maxwell, Kelvin–Voigt, Oldroyd and other models. One of the important non-Newtonian fluid is determined by the rheological relation (1 + λ d/dt)σ = 2ν(1 + κν −1 d/dt)E(v)

(1.2)

∑n where d/dt = ∂/∂t + i=1 vi ∂/∂xi is the total derivative and λ, κ, ν are positive constants. Fluids of (1.2) type have been introduced and extensively studied by Jeffreys and Oldroyd (see e.g. [12,15]). Integrating (1.2) along the velocity field v, solving it with respect to σ and substituting the result in Eq. (1.1) we get the following Jeffreys–Oldroyd initial–boundary value problem ∂v(t, x)/∂t + ∫

n ∑

vi (t, x)∂v(t, x)/∂xi − µ0 ∆v(t, x) −

(1.3)

i=1 t

exp((s − t)/λ) E(v)(s, z(s; t, x))ds +∇p(t, x) = f (t, x), (t, x) ∈ Q;

µ1 Div 0

div v(t, x) = 0, (t, x) ∈ Q; ∫ τ z(τ ; t, x) = x + v(s, z(s; t, x)) ds, 0 ⩽ t, τ ⩽ T,

(1.4) x ∈ Ω;

(1.5)

t

v(0, x) = v 0 (x), x ∈ Ω ;

v(t, x) = 0,

(t, x) ∈ Γ = [0, T ] × ∂Ω .

(1.6)

Here µ0 = 2κ, µ1 = 2(ν − κ) are constitutive constants. Details can be found in [26], Sec. 7.1. Survey of results on mathematical problems for Jeffreys–Oldroyd models is given in [22]. The integral term in (1.3) is related to the memory of the fluid along trajectories of the velocity field v. Note, that the system (1.3)–(1.6) contains not only unknown velocity v and pressure p, but also the trajectory z(τ ; t, x) being defined by the Cauchy problem (in the integral form) (1.5). Different models with memory have been studied in many papers (see. e.g. [1,4,5,10,13,14,16,22–26] et al.). Weak solvability of problem (1.3)–(1.6) where v in (1.5) is replaced by its smooth regularization v˜ have been considered in [23]. The reason of the regularization is the impossibility to define a unique classical solution to the Cauchy problem (1.5) for v from the class of weak solutions of problem (1.3)–(1.6). Relatively recent results on a solvability of Cauchy problem (1.5) for “low” regular v in the class of regular Lagrangian flows (generalization of the concept of a classical solution) allow to establish existence, uniqueness and stability of solutions to problem (1.5) in mentioned class (see e.g. [2,6–8]). This allows to get the existence of weak solutions to the problem (1.3)–(1.6) without a regularization of v in Eq. (1.5). For this purpose results of [2,6–8,23] have been substantially used. The paper is organized as follows. Section 2 provides basic notations, auxiliary statements and the statement of the main result. In Section 3 we introduce a family of two-parameterized regularized problems for (1.3)–(1.6). In Section 4 we prove the solvability of regularized problems. For this we reformulate these problems in the form of operator equations (Section 4.1) and use the topological degree theory for the solvability of these equations (Section 4.3). The solvability of regularized problems (1.3)–(1.6) is established in Section 4.4. In Section 5 we obtain estimates of solutions of regularized problems. Section 6 is devoted to the proof of the main Theorem 2.3. For this we construct a family of approximative regularized problems (Section 6.1) and on the base of results of Sections 4 and 5 we establish the solvability of approximative

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problems and convergence of their solutions to a limit function (Section 6.2). In Sections 6.3–6.4 we prove the fact that the limit function satisfies an appropriate integral identity and belongs to a functional class defining a solution of problem (1.3)–(1.6). Section 6.5 concludes the proof of the main Theorem. The constants in inequalities and in chains of inequalities which are independent of essential parameters are identified by a single letter M . 2. Auxiliary statements and main result We use the standard notations Lp (Ω ), Lp (Q), Wp1 (Ω ), Wpk,m (Q), 1 ≤ p ≤ +∞, for Banach spaces of functions on Ω and Q. By upper index n we denote Rn -valued function spaces while the sign n×n denotes the space of n × n matrix-valued functions. By C0∞ (Ω ) we denote the space of smooth compactly supported functions in Ω . Let V ={u : u ∈ C0∞ (Ω )n , div u = 0}. The Hilbert spaces H and V are defined as the completion of V in L2 (Ω )n and W21 (Ω )n , respectively (see e.g. [17]). Let ⟨f, v⟩ be the action of a functional f from dual to V space V ∗ on a function v from V . The sign ⟨f, φ⟩ denotes the action of a functional f ∈ V ∗ on an element φ ∈ V . By (·, ·) we denote the scalar products in Hilbert spaces V, L2 (Ω )n and L2 (Ω )n×n (it is clear from a context in which one). Sometimes we supply scalar products with correspondent indexes. We will use brief designations for norms: ∥v∥L2 (Ω)r = |v|0 , ∥v∥W k (Ω)r = |v|k for k = ±1, ∥v∥H = |v|0 , 2 ∥v∥V = |v|1 , ∥v∥L2 (0,T ;V ) = ∥v∥0,1 , ∥v∥L2 (0,T ;V ∗ ) = ∥v∥0,−1 , ∥v∥L2 (0,T ;H) = ∥v∥0 , ∥v∥L2 (0,T ;L2 (Ω)r ) = ∥v∥0 . Here r = 1, 2, 3 or n × n. Next, we use following notations for functional spaces: E = L2 (0, T ; V ) with the norm ∥v∥E = ∥v∥L2 (0,T ;V ) for v ∈ E; E ∗ = L2 (0, T ; V ∗ ) with the norm ∥f ∥E ∗ = ∥f ∥L2 (0,T ;V ∗ ) for f ∈ E ∗ ; E1 = L1 (0, T ; V ∗ ) with the norm ∥f ∥E1 = ∥f ∥L1 (0,T ;V ∗ ) for f ∈ E1 ; W ={v : v ∈ E, v ′ ∈ E ∗ } with the norm ∥v∥W =∥v∥E + ∥v ′ ∥E ∗ for v ∈ W ; W1 = {v : v ∈ E, v ′ ∈ E1 } with the norm ∥v∥W1 =∥v∥E + ∥v ′ ∥E1 for v ∈ W1 ; W2 = {v : v ∈ E, v ′ ∈ L4/3 (0, T ; V ∗ )} with the norm ∥v∥W2 =∥v∥E + ∥v ′ ∥L4/3 (0,T ;V ∗ ) for v ∈ W2 . Let W0 = W for n = 2 and W0 = W2 for n = 3. Let us consider the set C 1 D(Ω ) of one-to-one maps z : Ω → Ω coinciding with the identity map on ∂Ω and having continuous first order partial derivatives such that det(∂z/∂x) = 1 at each point of Ω . Assume that in this set the norm of the space of continuous functions C(Ω )n is used. Let us introduce the set CG = C([0, T ] × [0, T ], C 1 D(Ω )). Note, that CG ⊂ C([0, T ] × [0, T ], C 1 (Ω )n ), so below CG is regarded as a metric space with the metric defined by the norm of the space C([0, T ] × [0, T ], C(Ω )n ). Consider the Cauchy problem (1.5). In the case of v ∈ L1 (0, T ; C 1 (Ω )) with zero boundary condition problem (1.5) is non-locally uniquely solvable in the classical sense (see [14]). However, in the case of only integrable vector functions v the situation is more complicated and it is necessary to use a more general concept of solution to problem (1.5). Definition 2.1. A function z(τ ; t, x), (τ ; t, x) ∈ [0, T ] × [0, T ] × Ω is called Regular Lagrangian Flow (RLF ) associated to v if it satisfies the following conditions: (1) for a.e. x ∈ Ω and any t ∈ [0, T ] the function γ(τ ) = z(τ ; t, x) is absolutely continuous solution to Eq. (1.5); (2) for every t, τ ∈ [0, T ] and arbitrary Lebesgue measurable set B with measure m(B) the relation m(z(τ ; t, B)) = m(B) is fulfilled; (3) for all ti , ∈ [0, T ], i = 1, 2, 3, and a.e. x ∈ Ω the relation z(t3 ; t1 , x) = z(t3 ; t2 , z(t2 ; t1 , x)) holds. One can find the definition of RLF , for example, in [2,7,8]. Here we give the definition in the particular case of a bounded domain Ω and divergence free function v.

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Let us introduce some results on RLF . Let D = [0, T ] × [0, T ] and let L be the set of measurable on Ω functions which is considered as a metric space with the metric ∫ d(f, g) = |f (t, x) − g(t, x)|(1 + |f (t, x) − g(t, x)|)−1 dt dx. Q

Let vx be the Jacobi matrix of a vector function v. Theorem 2.1 ([8]). Let v ∈ L1 (0, T ; Wp1 (Ω )n ), 1 ≤ p ≤ +∞, div v(t, x) = 0 and v|[0,T ]×∂Ω = 0. Then there exists a unique RLF z ∈ C(D; Ln ) associated to v. Moreover, z(s; t, ·) ∈ W11 (Ω ), ∂ z(τ ; t, x) = v(τ, z(τ ; t, x)), t, τ ∈ [0, T ], a.e. x ∈ Ω , (2.1) ∂τ z(τ ; t, Ω ) ⊂ Ω

(up to am -negligible set).

(2.2)

◦

Theorem 2.2. Let divergence free functions v, v m belong to L1 (0, T ; Wp1 (Ω )n ), m = 1, 2, . . ., for some p > 1 and let the inequalities ∥vx ∥L1 (0,T ;Lp (Ω)n×n ) + ∥v∥L1 (0,T ;L1 (Ω)n ) ⩽ M, ∥vxm ∥L1 (0,T ;Lp (Ω)n×n ) + ∥v m ∥L1 (0,T ;L1 (Ω)n ) ⩽ M

(2.3)

hold. Let v m converges to v in L1 (Q)m as m → +∞. Let z m (τ ; t, x) and z(τ ; t, x) be RLF ′ s associated to v m and v, respectively. Then the sequence z m converges to z with respect to the Lebesgue measure in the set [0, T ] × Ω uniformly on t ∈ [0, T ]. In more general form this result is proved in [7], Corollaries 3.6, 3.7, 3.9. Let us note that from Theorem 2.1 it follows that Eq. (1.5) has a unique solution z(τ ; t, x) ≡ Z0 (v) in the class of RLF for every v ∈ W0 . Definition 2.2. Let f ∈ L2 (0, T ; V ∗ ) and v 0 ∈ H. A weak solution of (1.3)–(1.6) is a function v ∈ W0 satisfying the initial condition (1.6) and the identity n ∑ d (v, φ) − (vi v, ∂φ/∂xi ) + µ0 (E(v), E(φ)) + dt i=1 ∫ t µ1 exp((s − t)/λ)(E(v)(s, z(s; t, x)), E(φ)(x)) ds = ⟨f, φ⟩ (2.4) 0

for any φ ∈ V and a.e. t ∈ [0, T ]. Remark 2.1. E(φ))L2 (Ω)n×n .

Here (v, φ) = (v, φ)H , (vi v, ∂φ/∂xi ) = (vi v, ∂φ/∂xi )L2 (Ω)n , (E(v), E(φ)) = (E(v),

Remark 2.2. It is known that W ⊂ C([0, T ], H) and W1 ⊂ Cweak ([0, T ], H) (see [17, Ch. III, Lemmas 1.1, 1.2 and 1.4]). Consequently, since W0 = W for n = 2 and W0 = W2 for n = 3, the initial condition in (1.6) has a sense. Let us formulate the main result. Theorem 2.3. Let f ∈ L2 (0, T ; V ∗ ) and v 0 ∈ H. Then there exists a weak solution to problem (1.3)–(1.6). For the proof of Theorem 2.3 we construct a two-parameter family of more regular approximative problems.

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3. Approximative problems Let us make modifications in Eq. (1.3), so that all the terms will belong to L2 (0, T ; V ∗ ). We introduce the following auxiliary operators: ∑n 2 (1) Kε (v) = i=1 ∂(vi v(1 + ε|v| )−1 )/∂xi , ε ≥ 0, (2) a regularization Sδ : H → C 1 (Ω )n ∩ V for δ > 0 such that Sδ (v) → v in H as δ → 0 and the associated to Sδ map Sδ : L2 (0, T ; H) → L2 (0, T ; C 1 (Ω ) ∩ V ) which is continuous. One can find different types of the regularization operator in [13,26]. Recall the following properties of Sδ (see e.g. [26], Section 7.7): ∥Sδ ∥H→H ≤ M,

∥Sδ ∥H→C 1 (Ω) ≤ M, δ > 0;

|Sδ v − v|0 → 0 as δ → 0 for any v ∈ H.

(3.1) (3.2)

Consider the family of approximation problems depending on ε > 0 and δ > 0: ∂v/∂t +

n ∑

2

∂(vi v(1 + ε|v| )−1 )/∂xi − µ0 ∆v −

i=1

∫ t µ1 Div exp((s − t)/λ)E(v)(s, Zδ (v)(s; t, x)) ds + ∇p = f, (t, x) ∈ Q;

(3.3)

0

div v = 0,

x ∈ Ω;

(3.4)

v(0, x) = v 0 (x).

(3.5)

Here Zδ (v)(τ ; t, x) = z(τ ; t, x), where z = z(τ ; t, x) is the solution to the Cauchy problem ∫ τ z(τ ; t, x) = x + Sδ (v)(s, z(s; t, x)) ds, 0 ⩽ τ ; t ⩽ T, x ∈ Ω .

(3.6)

t

Definition 3.1. A function v ∈ W is called a weak solution of the regularized problem (3.3)–(3.5) if it satisfies the identity n ∑ d 2 (v, φ) − (vi v(1 + ε|v| )−1 , ∂φ/∂xi ) + µ0 (E(v), E(φ)) + dt i=1 ∫ t µ1 exp((s − t)/λ)(E(v)(s, Zδ (v)(s; t, x)), E(φ)(x)) ds = ⟨f, φ⟩

(3.7)

0

for any φ ∈ V and a.e. t ∈ [0, T ] and the initial condition (3.5). In order to reformulate the problem (3.3)–(3.5) in a more convenient operator form we introduce maps associated to the terms of Eq. (3.3): • a map A : V → V ∗ , ⟨A(u), h⟩ = µ0 (E(u), E(h)),

u, h ∈ V ;

(3.8)

Kε : V → V ∗ , ⟨Kε (v), h⟩ = n ∑ 2 ((vi v(1 + ε|v| )−1 ), ∂h/∂xi ), u, h ∈ V, ε ≥ 0;

(3.9)

• a map

i=1

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• for v ∈ V and z ∈ CG a functional C(v, z)(t) on V for every fixed t ∈ (0, T ): ⟨C(v, z)(t), h⟩ = ∫ t µ1 ( exp((s − t)/λ)E(v)(s, z(s; t, x)) ds, E(h)), h ∈ V.

(3.10)

0

Below we show that C : E × CG → E ∗ . Now, Definition 3.1 of a weak solution to the regularized problem (3.3)–(3.5) can be reformulated as follows. Definition 3.2. Given ε > 0, f ∈ L2 (0, T ; V ∗ ) and v 0 ∈ H, a weak solution to the regularized problem (3.3)–(3.5) is a function v ∈ W satisfying relations v ′ + A(v) − Kε (v) + C(v, Zδ (v)) = f, v(0) = v 0 .

(3.11)

Theorem 3.1. For any ε > 0, f ∈ L2 (0, T ; V ∗ ), and v 0 ∈ H problem (3.11) has at least one solution v ∈ W.

4. Proof of Theorem 3.1 4.1. Equivalent equations We introduce maps L, G and Nε :W → E ∗ × H by the equalities L(v) = (v ′ + A(v), v|t=0 ), G(v) = (C(v, Zδ (v)), 0) and Nε (v) = (Kε (v), 0). Then the problem (3.11) is equivalent to the operator equation in the Banach space W L(v) = Nε (v) − G(v) + ⟨f, v 0 ⟩.

(4.1)

Theorem 4.1. For any ε > 0, f ∈ L2 (0, T ; V ∗ ) and v 0 ∈ H Eq. (4.1) has solution v ∈ W and, hence, the problem (3.11) has at least one weak solution. To prove this we establish some auxiliary facts. Lemma 4.1. The following assertions hold: (1) the map A : E → E ∗ is continuous and satisfies the estimate ∥A(v)∥E ∗ ≤ M0 (1 + ∥v∥E );

(4.2)

(2) the relations K0 (v) ∈ E1 and Kε (v) ∈ L∞ (0, T ; V ∗ ) for v ∈ E and ε > 0 hold; (3) the maps K0 : E → E1 and Kε : E → E ∗ , ε > 0 are continuous and estimates ∥Kε (v)∥E ∗ ≤ M/ε, ε > 0,

∥Kε (v)∥L1 (0,T ;V ∗ ) ≤ M ∥v∥2E , ε ≥ 0

(4.3)

are valid; (4) the map Kε : W → E ∗ is compact for ε > 0. These facts can be found in [10] (Lemma 2.1 and Theorem 2.2). Let us note that W ⊂ E ∩ C([0, T ], H) (see [17, Ch. III, Lemma 1.2]). For v ∈ E ∩ C([0, T ], H) we introduce the norm ∥v∥EC = max0≤t≤T ∥v(t)∥H + ∥v∥E and the equivalent norms ∥v∥k,EC = ∥¯ v ∥EC , where v¯(t) = exp(−kt)v(t), k ≥ 0. Similarly, we define equivalent norms ∥ · ∥k,E , ∥ · ∥k,E ∗ ×H , ∥ · ∥k,L2 (Q) in spaces E, E ∗ × H and L2 (Q). The following Theorem is a particular case of Theorem 2.1 in [10].

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Theorem 4.2. The map L : W → E ∗ × H is invertible and for all functions u, v ∈ W and k ≥ 0 the estimate ∥v − u∥k,EC ≤ M ∥L(v) − L(u)∥k,E ∗ ×H

(4.4)

holds, where constant M is independent on u, v and k.

4.2. Properties of operator C Let us study properties of operator C. Lemma 4.2. For any v ∈ E and z ∈ CG the inclusion C(v, z) ∈ E ∗ holds and the map C : E × CG → E ∗ is continuous and bounded. Proof . By the definition of the map C we have for v ∈ E, z ∈ CG, h ∈ E and t ∈ (0, T ) ⟨C(v(t), z(·; t, ·)), h(t)⟩ = ∫ t µ1 ( exp((s − t)/λ)E(v)(s, z(s; t, x)) ds, E(h(t))). 0

First, establish the continuity and boundedness of the map ∫ t B : (v, z) ↦→ exp(−(t − s)/λ)E(s, z(s; t, x)) ds 0

from the space E × CG to the space L2 ([0, T ] × [0, T ]; L2 (Ω )n×n ). This map is a superposition of the integral operator and map Φ : (v, z) ↦→ E(v)(s, z(s; t, x))

(4.5)

n×n

from the space E × CG to the space L2 ([0, T ] × [0, T ], L2 (Ω ) ). For any fixed z ∈ CG we have ∫ T∫ T ∥E(v)(s, z(s; t, ·)) − E(u)(s, z(s; t, ·))∥2L2 (Ω)n×n ds dt = 0 0 ∫ T∫ T∫ E 2 (v − u)(s, z(s; t, x)) dx ds dt = 0 0 Ω ∫ T∫ T E 2 (v − u)(s, z) dz ds = 0 Ω ∫ T T ∥E(v − u)(s, ·)∥2L2 (Ω)n×n ds = T ∥v − u∥2E .

(4.6)

0

Note that the change of variable x by z = z(s; t, x) for fixed s, t does not change L2 (Ω ) norm since det(∂z/∂x) = 1 for all (s, t, x) ∈ [0, T ] × [0, T ] × Ω . Therefore, the map Φ is continuous with respect to the variable v uniformly with respect to z. Thus, it is enough now to prove the continuity of Φ with respect to the variable z for a fixed value v. Let zl be any sequence from CG, converging to z0 ∈ CG w.r.t. the norm C([0, T ] × [0, T ], C(Ω )n ) and ε > 0 be an n×n arbitrary number. Since the is dense in L2 (Q)n×n , there exists√a function √ set of continuous functions C(Q) ξ which is continuous ε/(3 T ) approximation of the function E(v), i.e. ∥E(v)−ξ∥L2 (Q)n×n < ε/(3 T ). Then, keeping designations of variables in the norms we have ∥E(v)(s, zl (s; t, x)) − E(v)(s, z0 (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) ≤ ∥E(v)(s, zl (s; t, x)) − ξ(s, zl (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) + ∥ξ(s, zl (s; t, x)) − ξ(s, z0 (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) + ∥E(v)(s, z0 (s; t, x)) − ξ(s, z0 (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) .

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By the choice of ξ the first and last terms here are less than ε/3. Function ξ is uniformly continuous on Q, therefore, the composition operator z ↦→ ξ(·, z) of CG to C([0, T ] × [0, T ], C(Ω )n×n ) is continuous and, therefore, ∥ξ(s, zl (s; t, x))−ξ(s, z0 (s; t, x))∥C([0,T ]×[0,T ],C(Ω)n×n ) → 0 as l → ∞. Choosing l > l0 large enough we find that ∥ξ(s, zl (s; t, x)) − ξ(s, z0 (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) < ε/3 and, hence, ∥E(v)(s, zl (s; t, x)) − E(v)(s, z0 (s; t, x))∥L2 ([0,T ]×[0,T ],L2 (Ω)n×n ) < ε. Thus, the continuity of Φ w.r.t. variable z is established. Estimate (4.6) for u = 0 and boundedness of the integral operator provide action and boundedness of the map B and, consequently, the continuity and the boundedness of the map C. Lemma 4.2 is proved. □ Lemma 4.3. The map Zδ : W1 → CG is continuous and for every weakly convergent sequence {vl }, vl ∈ W1 , vl ⇀ v0 there exists a subsequence {vlk } such that Zδ (vlk ) → Zδ (v0 ) w.r.t. the norm of C([0, T ] × [0, T ], C(Ω )n ). Proof . Since the embedding V ⊂ H is compact (see. [17], Ch.II, Theorem 1.1) and embedding W1 in L2 (0, T ; H) is compact (see [17], Ch. III, Remark 2.1.), there exists a subsequence {vlk } converging in L2 (0, T ; H) to v0 . By assumption, the operator Sδ is continuous in L2 (0, T ; H), so Sδ (vlk ) → Sδ (v0 ) strongly in L2 (0, T ; C 1 (Ω )n ∩ V ). Consider Eq. (1.5) on the set L2 (0, T ; C 1 (Ω )n ∩ V ). It has the solving operator Z0 . Show that Z0 acts from L2 (0, T ; C 1 (Ω )n ∩ V ) to CG and is continuous. Choose an arbitrary sequence {vl }, vl ∈ L2 (0, T ; C 1 (Ω )n ∩ V ), vl → v0 as l → ∞. Due to results of [14] for each v = vl , l = 1, 2, . . ., there exists a unique solution zl = Z0 (vl ) of Eq. (1.5) and the estimate ∫ τ ∥zl (τ ; t, ·) − z0 (τ ; t, ·)∥C(Ω)n ≤ M | ∥vl (s; ·) − v0 (s; ·)∥C(Ω)n ds| t

is valid for some independent on l, τ, t constant M and, consequently, ∥zl − z0 ∥C([0,T ]×[0,T ],C(Ω)n ) ≤ M ∥vl − v0 ∥L1 (0,T ;C 1 (Ω)n ) ≤ M ∥vl − v0 ∥L2 (0,T ;C 1 (Ω)n ) . Then the assumption vl → v0 as l → ∞ w.r.t. the norm of L2 (0, T ; C 1 (Ω )n ) implies that zl = Z0 (vl ) → Z0 (v0 ) = z0 w.r.t. the norm of C([0, T ] × [0, T ], C(Ω )n ). Thus the map Zδ is continuous as a superposition of the continuous operators Sδ and Z0 . Lemma 4.3 is proved. □ Lemma 4.4. For any fixed function z ∈ CG and arbitrary u, v ∈ E the estimate ∥C(v, z) − C(u, z)∥k,E ∗ ≤ δ1

√

T /(2k) ∥u − v∥k,E

holds. Proof . Let v(t, x) = exp(−kt)v(t, x) and u(t, x) = exp(−kt)u(t, x). By definition we have for h ∈ E ⟨exp(−kt)C(v, z) − exp(−kt)C(u, z), h⟩ = N ∫ T ∫ ∫ t ∑ µ1 exp(s − t)(1/λ + k)Eij (v − u)(s, z(s; t, x)) ds Eij (h)(t, x) dx dt. i,j=1

0

Ω

0

(4.7)

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Here ⟨f, φ⟩ means the action of f ∈ E ∗ on φ ∈ E. Using H¨older inequality and supposing (1/λ + k) = r we obtain |⟨exp(−kt)C(v, z) − exp(−kt)C(u, z), h⟩| ≤ ∫ T∫ t exp((s − t)r)|E 2 (v − u)(s, z(s; t, ·))|1 |E 2 (h)(t, ·)|1 ds dt = µ1 0

0

∫

T

∫

µ1

(∫ )1/2 exp((s − t)r) E 2 (v − u)(s, z(s; t, x)) dx × Ω

0

0

(∫

t

E 2 (h)(t, x) dx

)1/2 ds dt =

Ω

∫

T

µ1 0

∫ µ1 0

∫

T

(∫ )1/2 ∫ t 2 E (v − u)(s, z) dz exp((s − t)r) ∥h(t, ·)∥V ds dt = Ω 0 ∫ t exp((s − t)r)∥(v − u)(s, ·)∥V ∥h(t, ·)∥V ds dt ≤

0 T(∫ t

µ1 0

)1/2 (∫ t )1/2 exp(2r(s − t)) ds ∥(v − u)(s, ·)∥2V ds ∥h(t, ·)∥V dt ≤

0

0

(∫

T

∫

µ1 ∥v − u∥E ∥h∥E 0

t

)1/2 exp(2r(s − t)) ds dt .

0

Here E 2 (w) = (E(w), E(w)). From this and easily checked inequality ∫ T ∫ T∫ t 1 dt = T /(2(1/λ + k)) exp(2r(s − t)) ds dt ≤ 2(1/λ + k) 0 0 0 we have |⟨exp(−kt)C(v, z) − exp(−kt)C(u, z), h⟩| ≤ µ1 (T λ/(2 + 2λk))1/2 ∥v − u∥E ∥h∥E . From this the required estimate ∥C(v, z) − C(u, z)∥k,E ∗ = ∥ exp(−kt)(C(v, z) − C(u, z))∥E ∗ ≤ √ √ µ1 T λ/(2 + 2λk) ∥v − u∥E ≤ µ1 T /(2k) ∥v − u∥k,E follows. Lemma 4.3 is proved.

□

These statements allow us to study properties of the map G. Let γk be the Kuratowski measure of noncompactness (MNC) (see [3]) in the space E ∗ with the norm ∥ · ∥k,E ∗ . Recall the definition of a measure of noncompactness. Definition 4.1. A non-negative real function ψ, defined on subsets of a Banach space is called measure of noncompactness (MNC), if for any subset M of this space (1) ψ(co M) = ψ(M ), (2) for any two sets M1 , M2 such that M1 ⊂ M2 the inequality ψ(M1 ) ≤ ψ(M2 ) is valid. Here co M is the closure of the convex hull of the set M. As an example of M N C let us mention the Kuratowski M N C: the greatest lower bound of d > 0 for which the set M can be covered by a finite number of sets of diameter d. Kuratowski M N C and many others important examples of M N C have other important properties:

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(3) ψ(M) = 0 if M is pre-compact; (4) ψ(M ∪ K) = ψ(M), if K is pre-compact. Let X be a bounded subset of a Banach space and A : X → F be an arbitrary map of X into a Banach space F . The continuous bounded map g : X → F is called A-condensing if ψ(g(M)) ≤ ψ(A(M)) for any set M ⊆ X such that ψ(g(M)) ̸= 0. Here ψ is some M N C with the properties (1)–(3). Theorem 4.3. Map G : W → E ∗ × H is L-condensing with respect to Kuratowski M N C γk for sufficiently large k. Proof . Let M ⊂ W be an arbitrary bounded set. In view of Lemma 4.3 set Zδ (M) is pre-compact. Then the set C(v, Zδ (M)) is pre-compact for any fixed v ∈ W. In addition, for every z ∈ Zδ (M) map C(·, z) satisfies √ the Lipschitz condition with respect to the norms ∥ · ∥k,E and ∥ · ∥k,E ∗ with constant µ1 T /(2k). Then by √ Theorem 1.5.7 from [3] the map C(v, Zδ (v)) and, hence, the map G are µ1 T /(2k)-bounded with respect √ to the Hausdorff M N C χk , i.e. χk (G(M)) ≤ δ1 T /(2k) χk (M). It is well known (see. [3, Theorem 1.1.7]), that Hausdorff and Kuratowski M N C satisfy the inequalities χk (M) ≤ γk (M) ≤ 2χk (M). Then γk (G(M)) ≤ 2χk (G(M)) ≤ 2µ1

√

T /(2k) γk (M).

(4.8)

From estimate (4.4) and the definition of the Kuratowski M N C it follows that γk (M) ≤ M1 γk (L(M)).

(4.9)

Estimates (4.8) and (4.9) yield the estimate γk (G(M)) ≤ 2M1 µ1 Choosing k such that 2M1 µ1

√

√

T /(2k) γk (L(M)).

T /(2k) < 1 we obtain the assertion of Theorem 4.3. □

4.3. Solvability of Eq. (3.11) The solvability of the approximating differential-operator equation (3.11) is proved via the map degree theory (see. [9,20,21]). We introduce an auxiliary family of operator equations including the approximation Eq. (3.11): v ′ + A(v) − ξKε (v) + ξC(v, Zδ (v)) = f, v(0) = v 0 ,

ξ ∈ [0, 1], ε > 0.

(4.10)

For ξ = 1 (4.10) coincides with (3.11). Following the scheme of [10] let us establish a priori estimates for solutions of this family. Theorem 4.4. For any solution v ∈ W of problem (4.10) and ξ ∈ [0, 1] the estimates ∥v∥EC ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ),

∥v ′ ∥E ∗ ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2

(4.11)

are valid, with an independent on v and ξ ∈ [0, 1] but dependent, generally speaking, on ε > 0 constant M . Proof . Let v ∈ W be an arbitrary solution of problem (4.10) for some ξ ∈ [0, 1]. Then L(v) = ξNε (v) − ξG(v) + (f, v 0 ).

(4.12)

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Since L(0) = (0, 0) then from (4.4) it follows that ∥v∥k,EC ≤ M ∥L(v)∥k,E ∗ ×H .

(4.13)

Similarly, we have C(0, Zδ (v)) = 0 and estimate (4.7) implies the inequality ∥C(v, Zδ (v))∥k,E ∗ = ∥C(v, Zδ (v)) − C(0, Zδ (v))∥k,E ∗ ≤ √ µ1 T /(2k) ∥v∥k,E .

(4.14)

Taking into account estimate (4.3) we get from Eq. (4.12), estimate (4.13) and initial condition (4.14) that √ ∥v∥k,EC ≤ M (1/ε + δ1 T /(2k) ∥v∥k,E + ∥f ∥k,E ∗ + |v 0 |0 ). √ Since ∥v∥k,E ≤ ∥v∥k,EC , for sufficiently large k such that M δ1 T /(2k) < 1/2, we get ∥v∥k,EC ≤ 2M (1/ε + ∥f ∥k,E ∗ + |v 0 |0 ). Hence, taking into account the equivalence of the norms ∥ · ∥k,EC and ∥ · ∥EC , ∥·∥k,E ∗ and ∥·∥E ∗ we arrive to the first estimate (4.11). To prove the second estimate (4.11) it is sufficient to express v ′ from Eq. (4.10) and take advantage of estimates (4.11) and the boundedness of the maps A, Kε , C in E. Theorem 4.4 is proved. □

4.4. Conclusion of the proof of Theorem 3.1 Reduce the study of (3.11) to the study of equivalent operator equation (4.1). Consider a family of auxiliary problems (4.10) in an operator form L(v) − ξ(Nε (v) − G(v)) = (f, v 0 ).

(4.15)

By Theorem 4.3 and Lemma 4.1 the map ξ(Nε − G) of W × [0, 1] in E ∗ is L-condensing with respect to Kuratowski M N C γk . Moreover, from a priori estimates (4.11) it follows that equations (4.15) do not have solutions on the boundary of the ball BR ⊂ W of sufficiently large radius R with the center at zero. Therefore, for every ξ ∈ [0, 1] the topological degree of the map deg2 (L−ξ(Nε −G), B R , (f, v 0 )) (see e.g. [20]) is wellposed. Using the homotopy invariance of the topological degree w.r.t. ξ, we get deg2 (L − Nε + G, B R , (f, v 0 )) = deg2 (L, B R , (f, v 0 )). As L is invertible, the equation L(v) = (f, v 0 ) has a unique solution u0 in W and u0 satisfies a priori estimates (4.11) (changing v by u0 ). Then u0 ∈ BR and deg2 (L, B R , (f, v 0 )) = 1. Therefore, deg2 (L − Nε + G, B R , (f, v 0 )) = 1. The fact that the degree is not equal to zero yields the existence of solution of operator equation (4.15) or (4.1) and therefore the existence of a solution to problem (4.12) for ξ = 1 or (3.11). This proves Theorem 3.1. 5. Estimates of solutions to approximative problem (3.11) Next we show that the solutions of approximative problems (3.11) converge as ε → 0 to the solution of problem (1.3)–(1.6). For this we need a priori estimates of solutions of the regularized problem which are independent on ε. By virtue of [17, Lemma III.1.1 and Lemma III.1.4] function from W1 is weakly continuous on [0, T ] with values in H, so W1 ⊂ E ∩ L∞ (0, T ; H). Let us recall that above we have introduced for functions v ∈ E ∩ L∞ (0, T ; H) the norm ∥v∥EL = ∥v(t)∥L∞ (0,T ;H) + ∥v∥E and equivalent norms ∥v∥k,EL = ∥¯ v ∥EL , where v¯(t) = exp(−kt)v(t), k ≥ 0.

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Theorem 5.1. For any solution v ∈ W of problem (3.11) with ε > 0 the estimates ∥v∥EL ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ),

(5.1)

∥v ′ ∥L1 (0,T ;V ∗ ) ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2

(5.2)

are valid, with independent on ε constant M . Remark 5.1. Recall, that abbreviations of norms in right hand sides of inequalities are given in Section 2. Proof . Let v ∈ W be a solution of (3.11). Then v ′ + A(v) − Kε (v) + C(v, Zδ (v)) = f . Replace v(t) by expression ekt v(t). Multiplying these equations by e−kt , we obtain v ′ + kv + A(v) − K ε (v) + C(v, Zδ (ekt v)) = f ,

(5.3)

where K ε (v) = e−kt Kε (ekt v), C(v, Zδ (ekt v)) = e−kt C(ekt v, Zδ (ekt v)), f = e−kt f . The operator kv is defined by (kv, h) = k(v, h)L2 (Ω) for h ∈ V. Consider the action of functionals in the left-hand and right-hand sides of Eq. (5.3) upon a function v: 2

2

(1/2)(d/dt)|v(t)|0 + k|v(t)|0 + ⟨A(v(t)), v(t)⟩ − ⟨K ε (v(t)), v(t)⟩ = −⟨C(v, Zδ (ekt v))(t), v(t)⟩ + ⟨f , v(t)⟩. We know that (K ε (v(t)), v(t)) = 0 for all t ∈ [0, T ] (see [10]). Integrating both sides with respect to time on [0, t] we obtain 2

(1/2)|v(t)|0 + k∥v∥20 + µ0 ∥v∥20,1 = ∫ t ∫ t 2 ⟨f (τ ), v(τ )⟩ dτ. (1/2)|v 0 |0 − ⟨C(v, Zδ (ekτ v))(τ ), v(τ )⟩ dτ + 0

0

From here and estimate (4.7) for u = 0 in virtue of Cauchy inequality we obtain the inequality 2

(1/2)|v(t)|0 + k∥v∥20 + µ0 ∥v∥20,1 ≤ √ 2 (1/2)|v 0 |0 − µ1 T /(2k)∥v∥20,1 + ∥f ∥0,−1 ∥v∥0,1 . √ Assuming that k is large enough such that µ1 T /(2k) < µ0 /2 and using the Cauchy inequality, we arrive at the estimate ∥v∥2L∞ (0,T ;H) + 2k∥v∥20 + µ0 ∥v∥0,1 ≤ 2

2 |v 0 |0 + (1/2)∥v∥2L∞ (0,T ;H) + (1/2)µ0 ∥v∥20,1 + 2µ−1 0 ∥f ∥0,−1 ,

which implies the equivalent inequality (1/2)∥v∥2L∞ (0,T ;H) + 2k∥v∥20 + (1/2)µ0 ∥v∥0,1 ≤ 2

2 |v 0 |0 + 2µ−1 0 ∥f ∥0,−1

and the required estimate (5.1) follows. The proof of (5.2) is similar to the proof of the second estimate (4.11). From Eq. (3.11) we find that ′ v = f − A(v) + Kε (v) − C(v, Zδ (v)). Hence ∥v ′ ∥L1 (0,T ;V ∗ ) ≤ M (∥A(v)∥0,−1 + ∥Kε (v)∥L1 (0,T ;V ∗ ) + ∥C(v, Zδ (v))∥0,−1 + ∥f ∥0,−1 ).

(5.4)

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We estimate ∥Kε (v)∥L1 (0,T ;V ∗ ) as in [10]. For n ≤ 4 the embedding V ⊂ L4 (Ω )n is continuous. Then for u ∈ V by definition of Kε we have 2

∥Kε (u)∥V ∗ ≤ max ∥ui uj /(1 + ε|u| )∥L2 (Ω) ≤ max ∥ui uj ∥L2 (Ω) ≤ M ∥u∥2L4 (Ω)n . i,j

i,j

Therefore, ∥Kε (v)∥L1 (0,T ;V ∗ ) ≤ C∥v∥2L2 (0,T ;L4 (Ω)n ) . In force of the mentioned continuous embedding V ⊂ L4 (Ω )n the embedding E = L2 (0, T ; V ) ⊂ L2 (0, T ; L4 (Ω )n ) is also continuous. Then ∥v∥L2 (0,T ;L4 (Ω)n ) ≤ M ∥v∥0,1 and thus ∥Kε (v)∥L1 (0,T ;V ∗ ) ≤ M ∥v∥20,1 .

(5.5)

This estimate, boundedness of maps A and C on E and estimate (5.1) allow us to obtain the required estimate from inequality (5.4). This proves Theorem 5.1.

6. Proof of Theorem 2.3 6.1. Approximative problems Consider a depending on m = 1, 2, . . . family of regularized initial–boundary value problems (3.11) for ε = δ = 1/m (v m )′ + A(v m ) − K1/m (v m ) + C(v m , z m ) = f, v m (0) = v 0 . Here z m = z m (τ ; t, x) is a solution to the Cauchy problem ∫ τ z m (τ ; t, x) = x + v˜m (s, z m (s; t, x)) ds,

0 ≤ τ ; t ≤ T,

(6.1)

x∈Ω

t

where v˜m = S1/m v m . From Theorems 3.1 and 5.1 there follows the solvability of regularized problem (6.1) and corresponding estimates (5.1) and (5.2) for a solution v m ∈ W . From this, estimates (5.1) and (5.2) we have ∥v m ∥EL ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ),

(6.2)

∥(v m )′ ∥L1 (0,T ;V ∗ ) ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ).

(6.3)

We will show that v m (with up to a subsequence) converges to a solution to problem (1.3)–(1.6). 6.2. Passage to the limit Estimate (6.2) means that the sequence v m is bounded in L2 (0, T ; V ) and L∞ (0, T ; H). This implies the existence of function v ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H) such that v m (with up to a subsequence) converges to v weakly in L2 (0, T ; V ) and ∗-weakly in L∞ (0, T ; H). Using properties of lower limits of weakly and ∗-weakly converging sequences (see e.g. [19], Theorems 1 and 9, Chapter V, section 1) we get from estimates (6.2) and (6.3) the following inequality sup |v(t, ·)|0 + ∥v∥0,1 ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ).

0≤t≤T

(6.4)

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In addition, estimates (5.1) and (5.2) entail (see e.g. [17], Chapter III, proof of Theorem 3.2) the convergence of v m to v (with up to a subsequence) a.e. on [0, T ] × Ω . Let us show that the limit function v is a weak solution to problem (1.3)–(1.6). Obviously, function v m satisfies the identity n ∫ T ( ) ∑ 2 − vim (t, x)(1 + m−1 |v| )−1 v m (t, x), ∂φ(x)/∂xi ψ(t) dt 0

i=1

∫

T

(E(v m )(t, x), E(φ)(x))ψ(t) dt ∫ T ∫ t + µ1 ( exp((s − t)/λ)E(v m )(s, z m (s; t, x)) ds, E(φ)(x))ψ(t) dt 0 0 ∫ T ∫ T + (v m (t, x), φ)ψ ′ (t) dt = ⟨f (t, x), φ(x)⟩ψ(t) dt

+ µ0

0

0

(6.5)

0

for any φ ∈ V and ψ ∈ C0∞ (0, T ). From estimates (6.2) and (6.3) it follows (see e.g. [24]) that the sequence v m converges (with up to subsequence) to v weakly in L2 (0, T ; V ), *-weakly in L∞ (0, T ; H), strongly in L2 (Q)n , a.e. on Q = [0, T ]×Ω . Also the sequence of derivatives dv m /dt is bounded in the norm of the space L1 (0, T ; V ∗ ) and converges to dv/dt in the sense of distributions on [0, T ]. Next, we will need several Lemmas. Lemma 6.1. The sequence v˜m converges to v in L2 (0, T ; H). Proof of Lemma 6.1. It is easy to see that v˜m − v = I1 (m) + I2 (m), where I1 (m) = S1/m (v m − v), I2 (m) = S1/m v − v. From the first estimate (3.1) it follows that ∥S1/m ∥L2 (0,T ;H)→L2 (0,T ;H) ≤ M . Then ∥I1 (m)∥0 ≤ M ∥v − v m ∥0 → 0

as m → +∞.

For the proof that ∥I2 (m)∥0 → 0 as m → +∞ it suffices to show that ∫ T 2 |S1/m v(t, ·) − v(t, ·)|0 dt → 0 as m → +∞.

(6.6)

0

In virtue of (3.2) we have for integrand in (6.6) g(t) ≡ |˜ v m (t, ·) − v(t, ·)|0 = |S1/m v(t, ·) − v(t, ·)|0 → 0

as m → +∞.

In addition, on the strength of the first inequality in (3.1) function g 2 (t) satisfies the inequality g 2 (t) ≤ 2 M |v(t, ·)|0 . Hence, in virtue of the Lebesgue Theorem ∥I2 (m)∥0 → 0 as m → +∞. The assertion of Lemma 6.1 follows from the fact that ∥Ik (m)∥0 → 0 as m → +∞ for k = 1, 2. □ Consider the Cauchy problem (1.5) for the limit function v. Since v ∈ W , we have that v satisfies conditions of Theorem 2.1. Then from [2], Theorem 2.1, the existence of associated to v RLF z(τ ; t, x) in [0, T ] × [0, T ] × Ω follows. Lemma 6.2. The sequence z m (τ ; t, x) converges in (τ, x) Lebesgue measure on [0, T ] × Ω to z(τ ; t, x) for t ∈ [0, T ].

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Proof of Lemma 6.2. By the second estimate in (3.1) we have the uniform inequality ∥˜ vxm ∥L1 (Ω) ≤ M |˜ vxm |0 ≤ M |S1/m v m (t, ·)|1 ≤ M |v(t, ·)|1 . Hence, from estimates (5.1) there follows a uniform on m estimate of norms ∥v m ∥0,1 and, moreover, of norms ∥vxm ∥L1 (0,T ;L1 (Ω)) and ∥v m ∥L1 (0,T ;L1 (Ω)) . Using now Theorem 2.2 we get the assertion of Lemma 6.2. □ Next, the same estimates (5.1) and Theorem 2.2 imply that the sequence of RLF ′ s z m (τ ; t, x) converges (with up to subsequence) in [0, T ] × Ω for t ∈ [0, T ] as a function of variables (τ, x) ∈ [0, T ] × Ω to RLF z(τ ; t, x) associated to v ∈ L2 (0, T ; V ). By virtue of Lemma VI.5.1, [18] from the statement of Lemma 6.2 it follows that the sequence z m (τ ; t, x) converges (with up to subsequence) to z(τ ; t, x) a.e. on the set (τ, x) ∈ [0, T ] × Ω . 6.3. Proof of identity (2.4) for the limit function v Let us prove an auxiliary integral identity for v. Lemma 6.3. The limit function v(t, x) satisfies the identity ∫ T n ∫ T ∑ ′ (v(t, x), φ)ψ (t) dt − (vi (t, x)v(t, x), ∂φ(x)/∂xi t)ψ(t) dt 0

i=1

0

T

∫

(E(v)(t, x), E(φ)(x))ψ(t) dt ∫ T ∫ t + µ1 ( exp((s − t)/λ)E(v)(s, z(s; t, x)) ds, E(φ)(x))ψ(t) dt 0 0 ∫ T = ⟨f (t, x), φ(x)⟩ψ(t) dt + µ0

0

for any φ ∈ V and ψ ∈

(6.7)

0 C0∞ (0, T ).

Proof of Lemma 6.3. First, let φ ∈ V be smooth. Introduce notations for terms in the left hand side of (6.5): ∫ T J1m = (v m (t, x), φ)ψ ′ (t) dt, 0 n ∫ ∑

T

2

(vim (t, x)(1 + m−1 |v m | )−1 v m (t, x), ∂φ(x)/∂xi )ψ(t) dt, ∫ T i=1 m J3 = µ0 (E(v m )(t, x), E(φ)(x))ψ(t) dt, ∫ T ∫ t 0 = µ1 ( exp((s − t)/λ)E(v m )(s, z m (s; t, x)) ds, E(φ)(x))ψ(t) dt.

J2m =

0

J4m

0

0

Let the corresponding terms in the left hand side of (6.7) be ∫ T n ∫ T ∑ J1 = (v(t, x), φ)ψ ′ (t) dt, J2 = (vi (t, x), ∂φ(x)/∂xi )ψ(t) dt, ∫ T 0 i=1 0 J3 = µ0 (E(v)(t, x), E(φ)(x))ψ(t) dt, 0

∫ J4 = µ1 0

T ∫ t ( exp((s − t)/λ)E(v)(s, z(s; t, x)) ds, E(φ)(x))ψ(t) dt. 0

Here z is the RLF associated to v.

88

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

Proposition 6.1. Jim converges to Ji for i = 1, 2, 3. Proof of Proposition 6.1. The weak convergence of v m to v in L2 (0, T ; V ) implies that Jim converges to Ji for i = 1, 3. 2 Next, the sequence of functions vim (t, x)v m (t, x)(1 + m−1 |v m | )−1 weakly converges to vi (t, x)v(t, x) as m → +∞ in L2 ([0, T ] × Ω ) (see [10]). Then in virtue of Lemma 2.2 from [10] there follows the convergence of J2m to J2 as m → +∞. Proposition 6.1 is proved. Consider the term J4m . Obviously, ∫ T ∫ t J4m = µ1 ( exp((s − t)/λ) × 0 0 ∫ E(v m )(s, z m (s; t, x)) : E(φ)(x) dx ds)ψ(t) dt,

(6.8)

Ω

∫ T ∫ t J4 = µ1 ( exp((s − t)/λ) × 0 0 ∫ E(v)(s, z(s; t, x)) : E(φ)(x) dx ds)ψ(t) dt.

(6.9)

Ω

We will show that limm→∞ J4m = J4 . The integrand in formula (6.9) contains the superposition of functions E(v) and z. Let us show the summability of this superposition. First, let us establish Proposition 6.2. Let h ∈ L1 (Ω ). Then h(z(s; t, ·)) ∈ L1 (Ω ) for s, t ∈ (0, T ] and ∥h(z(s; t, ·))∥L1 (Ω) = ∥h(·)∥L1 (Ω) .

(6.10)

Proof of Proposition 6.2. Let hn (x), n = 1, 2, . . . be a sequence of smooth functions on Ω such that ∥hn (x)−h(x)∥L1 (Ω) → 0 as n → +∞ and hn (x) → h(x) on a dense subset Ω0 ⊂ Ω . The function hn (z(s; t, x)) is measurable. Using the change of variable x = z(t; s, y) (y = z(s; t, x) is the reverse change) we obtain the equality ∫ ∫ |hn (z(s; t, x))| dx = |hn (y)| dy. (6.11) Ω

Ω

We will show that hn (z(s; t, x)) → h(z(s; t, x)) as n → +∞ a.e. on Ω . Let Ω1 = {x : x = z(t; s, y), y ∈ Ω0 }. The definition of RLF yields Ω 1 = Ω . By (1.5) we have for x ∈ Ω1 hn (z(s; t, x)) = hn (z(s; t, z(t; s, y))) = hn (y), y ∈ Ω0 . It follows that hn (z(s; t, x)) = hn (y) → h(y) = h(z(s; t, x)) on the dense set Ω1 . Thus, h(z(s; t, x)) is measurable. From (6.11) and the convergence of hn (x) to h(x) in L1 (Ω ) it follows that the sequence ∥hn (z(s; t, x))∥L1 (Ω) is uniformly bounded. Using the Fatough theorem for |hn (z(s; t, x))| we get the summability of h(z(s; t, x)). The relation (6.10) for a bounded function h(x) follows from (6.11). In the case of an arbitrary h(x) it suffices to take a sequence of truncation functions [h(z(s; t, x))]k for h(z(s; t, x)) and using the equality [h(z(s; t, x))]k = [h]k (z(s; t, x)), to obtain (6.10) by means of the passage to the limit as k → +∞. Recall that [h(x)]k = h(x) for |h(x)| ≤ k, [h(x)]k = k if h(x) > k and [h(x)]k = −k if h(x) < −k. Proposition 6.2 is proved.

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

89

The proof of the summability of vx (s, z(s; t, x)) as a function of three variables is similar to the proof of Proposition 6.2 but it consists of some tedious arguments and we skip it. Proposition 6.3. Ji4 converges to J4 . Proof of Proposition 6.3. It is easy to see that J4m − J4 = Z1m + Z2m ,

(6.12)

where ∫ T ∫ t exp((s − t)/λ) Z1m = µ1 ( 0 0 ∫ × (E(v m )(s, z m (s; t, x)) − E(v)(s, z m (s; t, x))) : E(φ)(x) dx ds) ψ(t) dt; Ω ∫ T ∫ t m Z2 = µ1 ( exp((s − t)/λ) 0 0 ∫ × [E(v)(s, z m (s; t, x)) − E(v)(s, z(s; t, x))] : E(φ)(x) dx ds)ψ(t)dt.

(6.13)

Ω

∑n Here A : B = i,j=1 aij bij for matrixes A and B with coefficients aij and bij . We will show that limm→∞ Z1m = 0. Denote the integral over Ω in Z1m by ∫ I= (E(v m )(s, z m (s; t, x)) − E(v)(s, z m (s; t, x))) : E(φ)(x) dx. Ω

Let us make in I the change of variable x = z m (t; s, y)

(the reverse change is y = z m (s; t, x)).

(6.14)

Then ∫ I=

[E(v m )(s, y) − E(v)(s, y)] : E(φ)(z m (t; s, y)) dy.

Ω

By changing the integration order in Z1m we have ∫ T∫ t Z1m = µ1 exp((s − t)/λ) ∫0 0 × (E(v m )(s, y) − E(v)(s, y)) : E(φ)(z m (t; s, y)) dy ds ψ(t) dt Ω ∫ T∫ T = µ1 exp((s − t)/λ) ∫0 s × (E(v m )(s, y) − E(v)(s, y)) : E(φ)(z m (t; s, y)) dy ψ(t) dt ds Ω m = µ1 Z12 .

(6.15)

It is obvious that ∫ T∫ T m Z12 = exp((s − t)/λ) ∫0 s × (E(v m )(s, y) − E(v)(s, y)) : (E(φ)(z m (t; s, y)) − E(φ)(z(t; s, y))) dy ψ(t) dt ds Ω ∫ T∫ T ∫ + exp((s − t)/λ) (E(v m )(s, y) − E(v)(s, y)) : E(φ)(z(t; s, y)) dy ψ(t) dt ds 0

s

Ω

m m = Z121 + Z122 .

The weak convergence of v

m

(6.16) to v in L2 (0, T ; V ) implies that

m Z122

→ 0 as m → +∞.

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

90

Using the boundedness of functions ψ and exp((s − t)/λ) and applying the Cauchy–Schwarz and H¨older inequalities we get ∫ T∫ T m 2 |Z121 exp((s − t)/λ) | ≤ M( s

0

× |v m (s, ·) − v(s, ·)|1 |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 ψ(t) dt ds)2 ∫ T ∫ T m ≤ M( |v (s, ·) − v(s, ·)|1 |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt ds)2 0 s ∫ T ∫ T ∫ T 2 m ≤M |v (s, ·) − v(s, ·)|1 ds ( |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt)2 ds 0 0 s ∫ T ∫ T m 2 ≤ M ∥v − v∥0,1 ( |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt)2 ds. 0

m Let us show that Z121 → 0 as m → +∞. Denote the last factor in (6.17) by ∫ T ∫ T Ψm (s) = ( |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt)2 ds. 0

(6.17)

s

(6.18)

s

Represent it in the form ∫ Ψm (s) =

T

∫ gm (s) ds, gm (s) = ( s

0

T

|φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt)2 .

Let us establish the convergence gm (s) → 0 as m → +∞ for all s ∈ [0, T ]. It is easy to see that ∫ T gm (s) = ( |φx (z m (t; s, ·)) − φx (z(t; s, ·))|0 dt)2 s ∫ T∫ =( |φx (z m (t; s, y)) − φx (z(t; s, y))| dy dt)2 . s

(6.19)

Ω

Let ε > 0 be a small number. The continuity of function φx in Ω implies the existence of δ1 (ε) > 0 such that if |x′′ − x′ | ≤ δ1 (ε), then |φx (x′′ ) − φx (x′ )| ≤ ε.

(6.20)

Since the sequence z m (t; s, y) converges to z(t; s, y) in (t, y) Lebesgue measure on [s, T ] × Ω for s ∈ [0, T ], then for δ1 (ε) there exists such N = N (δ1 (ε)) that for m ≥ N the following inequality holds m({(t, y) : |z(t, s, y) − z m (t, s, y)| ≥ δ1 (ε)}) ≤ ε.

(6.21)

Let us denote Q(> δ1 (ε)) = {(t, y) ∈ Q : |z(t, s, y) − z m (t, s, y)| > δ1 (ε)}, Q(≤ δ1 (ε)) = {(t, y) ∈ Q : |z(t, s, y) − z m (t, s, y)| ≤ δ1 (ε)}. Then from (6.19) it follows that gm (s) ≤ M (G3 + G4 ),

(6.22)

where ∫ G3 =

2

|φx (z(t, s, y)) − φx (z m (t, s, y))| dy dt,

Q(>δ1 (ε))

∫ G4 = Q(≤δ1 (ε))

2

|φx (z(t, s, y)) − φx (z m (t, s, y))| dy dt.

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

For G4 by (6.20) we have |z(t, s, y) − z m (t, s, y)| ≤ δ1 (ε), and, therefore, ∫ G4 ≤ ε2 dy dt ≤ M ε2 .

91

(6.23)

Q(≤δ1 (ε))

In virtue of (6.21) we have m(Q(> δ1 (ε))) ≤ ε. Then ∫ G3 ≤ 2∥φx ∥C(Ω)

dy dt ≤ 2ε∥φx ∥C(Ω) .

(6.24)

Q(>δ1 (ε))

From estimates (6.23), (6.24) and (6.22) it follows that for a small ε > 0 and m ≥ N (δ1 (ε)) the following inequality holds gm (s) ≤ M ε.

(6.25)

The convergence gm (s) → 0 as m → +∞ for all s ∈ [0, T ] is established. Moreover, gm (s) is bounded because φx (s) is smooth. Therefore, Ψm → 0. The first factor in the right hand side of (6.17) is bounded w.r.t. m due to the boundedness of ∥v m ∥0,1 and for the second factor the convergence Ψm → 0 as m → +∞ holds. m Thus, (6.17) and (6.18) imply Z121 → 0 as m → +∞. m The convergence Z122 → 0 as m → +∞ follows from the weak convergence of v m → v in L2 (0, T ; V ). m m m From convergences Z121 → 0 and Z122 → 0 as m → +∞ and (6.16) we get Z12 → 0 as m → +∞. Thus, we have proved that |Z1m | → 0 as m → +∞.

(6.26)

|Z2m | → 0 as m → +∞.

(6.27)

Show that

Let us estimate |Z2m |. Let us approximate v(t, x) by some smooth and finite on [0, T ] × Ω function v˜, so that ∥v − v˜∥0,1 ≤ ε2 for an arbitrarily small number ε2 > 0. Then m m m |Z2m | ≤ M (Z21 + Z22 + Z23 ),

(6.28)

where m Z21 =

T

∫ 0

m Z22 =

exp((s − t)/λ)|v(s, z m (s; t, x)) − v˜(s, z m (s; t, x))|1 ds dt,

0 T

∫ ∫

0

t

∫

0 m Z23 =

t

∫

0 T

∫ 0

exp((s − t)/λ)|˜ v (s, z m (s; t, x)) − v˜(s, z(s; t, x))|1 ds dt, t

exp((s − t)/λ)|˜ v (s, z(s; t, x)) − v(s, z(s; t, x))|1 ds dt.

m Making the change of variable (6.14) in the integral that defines the norm |·|1 in Z21 , we get

|v(s, z m (s; t, x)) − v˜(s, z m (s; t, x))|1 = |v(s, y) − v˜(s, y)|1 .

(6.29)

m Similarly, using the change of variable y = z(s; t, x) we get for Z23

|˜ v (s, z(s; t, x)) − v(s, z(s; t, x))|1 = |˜ v (s, y) − v(s, y)|1 .

(6.30)

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

92

From inequalities (6.29) and (6.30) it follows that ∫ T∫ t m m exp((s − t)/λ)|v(s, ·) − v˜(s, ·)|1 ds dt Z21 + Z23 ≤ M 0

0

≤ M ∥v − v˜∥0,1 ≤ M ε2 .

(6.31)

Since v˜ is compactly supported, we have ∫ T ∫ t ∫ m Z22 ≤M ( exp((s − t)/λ) |˜ vx (s, z m (s; t, ·)) − v˜x (s, z m (s; t, ·))| dx ds)dt. k1

0

Ω

Since z m (s; t, x) converges a.e. to z m (s; t, x) on [0, T ] × Ω uniformly with respect to t and the function v˜x (t, x) is smooth and bounded, we get the convergence (6.27) as m → +∞ by the Lebesgue theorem. The assertion of Proposition 6.3 follows from (6.26) and (6.27). Proposition 6.3 is proved. Thus, due to Propositions 6.1 and 6.3 it is possible to pass to the limit in each term in (6.5), that yields the identity (6.7) for any smooth φ. Let us establish identity (6.7) for arbitrary φ ∈ V and ψ ∈ C0∞ (0, T ). We rewrite (6.7) for a smooth φ in the form [G1 , φ] − [G2 , φ] = 0,

(6.32)

where T

∫

′

(v(t, x), φ)ψ (t) dt −

[G1 , φ] = 0

∫

n ∫ ∑ i=1

T

(vi (t, x)v(t, x), ∂φ(x)/∂xi )ψ(t) dt

0

T

(E(v)(t, x), E(φ)(x))ψ(t) dt ∫ T ∫ t + µ1 ( exp((s − t)/λ)E(v)(s, z(s; t, x)) ds, E(φ)(x))ψ(t) dt, 0 0 ∫ T [G2 , φ] = (f (t, x), φ(x))ψ(t) dt. + µ0

0

(6.33) (6.34)

0

Let us obtain estimates of [Gi , φ], i = 1, 2, for arbitrary φ ∈ V . Proposition 6.4. Let φ be smooth. Then |[G1 , φ]| ≤ M |φ|1 , |[G2 , φ]| ≤ M |φ|1 .

(6.35)

Proof of Proposition 6.4. We denote g(t) = −(f (t, ·), φ(·)) +

n ∑

(vi (t, ·)v(t, ·), ∂φ(·)/∂xi ) + µ0 (E(v)(t, ·), E(φ)(·))

i=1

(∫

t

) E(v)(s, z(s; t, ·))ds, E(φ)(·)

+ µ1

≡

0

4 ∑

Ri .

i=1

Here R1 = −(f (t, ·), φ(·)), R2 =

n ∑

(vi (t, ·)v(t, ·), ∂φ(·)/∂xi ) ,

i=1 (∫

R3 = µ0 (E(v)(t, ·), E(φ)(·)), R4 = µ1 0

t

) E(v)(s, z(s; t, ·))ds, E(φ)(·) .

(6.36)

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

Let us estimate Ri . It is easy to see that the following inequality holds ∫ T ∫ T |R1 (t)| dt ≤ |φ|1 |f (t, ·)|−1 dt ≤ M ∥f ∥0,−1 |φ|0 . 0

93

(6.37)

0

The continuous embedding W21 (Ω ) ⊂ L4 (Ω ) for n = 2, 3, inequalities 1/4

1/2

1/2

3/4

∥u∥L4 (Ω) ≤ 21/4 |u|0 |u|1 , n = 2, ∥u∥L4 (Ω) ≤ M |u|0 |u|1 , n = 3 (see [17, Lemma III.3.3 and III.3.5]) yields ∫ T n ∫ ∑ |R2 (t)| dt ≤ 0

i=1 T

∫ ≤M 0

T

0

3/2

|v(t, ·)|1

|vi (t, ·)v(t, ·)|0 |∂φ(·)/∂xi |0 dt 1/2

|v(t, ·)|0

dt |φ|1 .

Elementary calculations and estimate (6.4) yield ∫ T ∫ T 2 2 |R2 (t)| dt ≤ M (|v(t, ·)|1 + |v(t, ·)|0 ) dt |φ|1 0

0

≤

M (∥v∥20,1

2

+ sup |v(t, ·)|0 )| φ|1 ≤ M (1 + ∥f ∥0−1 + |v 0 |0 )2 | φ|1 .

Estimate (6.4) implies for R3 (t) ∫ T ∫ |R3 (t)| dt ≤ M 0

(6.38)

t

0

T

|v(t, ·)|1 dt|φ|1 ≤ M (1 + ∥f ∥0−1 + |v 0 |0 )|φ|1 .

(6.39)

Finally, by the preceding arguments we have for R4 (t) ∫ T ∫ T ∫ t |R4 (t)| dt ≤ M | exp((s − t)/λ)E(v)(s, z(s; t, x)) ds|0 |E(φ)|0 dt 0 0 0 ∫ T ∫ t ≤M | exp((s − t)/λ)E(v)(s, z(s; t, x)) ds|0 dt|φ|1 0 0 ∫ T∫ t ≤M exp((s − t)/λ)|v(s, z(s; t, x))|1 dt|φ|1 0

∫

0

T

(∫

≤M 0

0

t

)1/2 (∫ t )1/2 2 dt|φ|1 exp(2(s − t)/λ) ds |v(s, y)|1 ds 0

≤ M (1 + ∥f ∥0−1 + |v 0 |0 )|φ|1 .

(6.40)

Taking into account smoothness of ψ(t), relation (6.36) and estimates (6.37)-(6.40) we get (6.35). Proposition 6.4 is proved. Since the set of smooth functions is dense in V , for φ ∈ V there exists a sequence of smooth functions φl ∈ V such that |φl − φ|1 → 0 as l → +∞. By (6.32) for φ = φl we have [G1 , φ] − [G2 , φ] = [G1 , φ − φl ] − [G2 , φ − φl ] + [G1 , φl ] − [G2 , φl ] = [G1 , φ − φl ] − [G2 , φ − φl ]. This equality and (6.35) entail |[G1 , φ] − [G2 , φ]| ≤ M |φ − ϕl |1 .

(6.41)

Taking into account inequality (6.41) and passing to the limit as l → +∞ in (6.7) for φ = φl we get (6.32), or that is the same, the identity (6.7) for any φ ∈ V . □

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

94

In order to prove that v is a weak solution to problem (1.3)–(1.6) it is now enough to prove that v ∈ W and satisfies the identity (2.4). First, let us prove the second assertion. Lemma 6.4. The limit function v satisfies the identity (2.4). Proof of Lemma 6.4. We will show that (2.4) follows from (6.7). For this we use the following fact, proved in [17], Lemma 1.1, Chapter III. Proposition 6.5. Let X be a Banach space and u, g ∈ L1 (0, T ; X). Then the following statements are equivalent: (1) for any ψ ∈ C0∞ (0, T ) the identity holds ∫ T ∫ ′ u(t)ψ (t) dt = − 0

T

(ψ ′ = dψ/dt);

g(t)ψ(t) dt

0

(2) for each η ∈ X ′ (the space X ′ is adjoint to X) d⟨u, η⟩/dt = ⟨g, η⟩ takes place in the sense of distributions on (0, T ); (3) u(t) is differentiable by a.e. t ∈ [0, T ] as a function with values in X and du(t)/dt = g(t). We rewrite (6.7) in the form ∫

T

u(t)ψ ′ (t) dt =

0

∫

T

g(t)ψ(t) dt, 0

where u(t) = (v(t, ·), φ(·)), and g(t) is determined by the formula (6.36). To prove (2.4) it is enough to establish summability of scalar functions u(t) and g(t) on [0, T ] and to use the scalar version of Proposition 6.5. Show first that u(t) ∈ L1 (a, b). Indeed, using the Cauchy inequality and the estimate (6.4), we obtain ∫ T ∫ T ∫ T |u(t, ·)|0 dt = |(v(t, ·), φ(·))|0 dt ≤ |v(t, ·)|0 |φ(·)|0 dt 0

0

0

≤ M T sup |v(t, ·)|0 |φ|0 ≤ M (1 + ∥f ∥0−1 + |v 0 |0 )|φ|0 .

(6.42)

0≤t≤T

Thus, u ∈ L1 (0, T ). Let us establish summability of g(t). To do this, we will show that Ri ∈ L1 (0, T ). From inequalities (6.37) –(6.40) there follows the summability of Ri , i = 1, . . . , 4 on [0, T ]. From this fact the summability of g(t) follows, and, therefore, the validity of (2.4). □ To complete the proof of Theorem 2.3 it remains to show that v ∈ W0 . Lemma 6.5. The limit function v belongs to the space W0 .

6.4. Proof of Lemma 6.5 It is convenient to rewrite problem (1.3)–(1.6) in an operator form dv/dt + A(v) − K(v) + C(v, z) = f

(for a.e. t ∈ [0, T ])

in the Hilbert space V ∗ . Here operators A(v), K(v) = K0 (v) and C(v, z) are defined by (3.8)–(3.10). Let us establish properties of operators in (6.43).

(6.43)

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

95

Proposition 6.6. Let v ∈ L2 (0, T ; V ) ∩ L0 (0, T ; H). Then ∥A(v)∥0,−1 ≤ M ∥v∥0,1 ;

∥K(v)∥0,−1

(6.44)

∥C(v, z)(t)∥0,−1 ≤ M ∥v∥0,1 ; ∫ T 2 ≤ M sup |v(t, ·)|0 ( |v(t, ·)|1 dt)1/2 (for n = 2); t

(6.45) (6.46)

0

1/2

3/2

∥K(v)∥L4/3 (0,T ;V ∗ ) ≤ M sup |v(t, ·)|0 ∥v∥0,1 (for n = 3).

(6.47)

t

Proof of Proposition 6.6. It is easy to see that for fixed u ∈ V and for every h ∈ V |⟨A(u), h⟩| ≤ M |(E(u), E(h))L2 (Ω)n×n |0 ≤ M |u|1 |h|1 . Therefore, |A(u)|−1 ≤ M |u|1 . This easily yields (6.44). By the standard arguments we get for C ∫ t |C(v, z)(t)|−1 ≤ M | exp((s − t)λ)E(v)(s, z(s; t, x)) ds|0 0 ∫ t ≤M exp((s − t)λ)|vx (s, z(s; t, x))|0 ds.

(6.48)

0

Using the change of variable y = z(s; t, x), we have ∫ ∫ 2 2 2 2 I = |vx (s, z(s; t, x))|0 = |vx (s, z(s; t, x))| dx = |vx (s, y)| dy = |vx (s, ·)|0 . Ω

Ω

Hence from (6.48) using the change of variable τ = s − t we obtain ∫ t |C(v, z)(t)|−1 ≤ M | exp((s − t)λ)|vx (s, ·)|0 ds 0 ∫ t ∫ 0 ≤M exp((s − t)λ)|v(s, ·)|1 ds = M exp(λτ )|v(t + τ, ·)|1 dτ. 0

(6.49)

0

Then, we deduce from (6.49) by virtue of the Minkowski integral inequality ∫ 0 ∥C(v, z)(t)∥0,−1 ≤ M ∥ exp(λτ )|v(t + τ, ·)|1 dτ ∥L2 (0,T ] 0 ∫ 0 ≤M exp(λτ )∥ |v(t + τ, ·)|1 ∥L2 (0,T ] dτ ≤ M ∥v∥0,1 . 0

Inequality (6.45) is proved. 1/2 1/2 Consider K. Let n = 2. Using inequality ∥u∥L4 (Ω) ≤ M |u|0 |u|1 for u ∈ V we get n ∑ |K(v)|−1 ≤ M | vi v|0 ≤ M ∥v∥2L4 (Ω) ≤ M |v|0 |v|1 , v ∈ V. i=1

From inequality (6.50) it follows that ∫ T 2 2 2 |K(v)|−1 dt)1/2 ( |v(t, ·)|0 |v(t, ·)|1 dt)1/2 0 0 ∫ T 2 |v(t, ·)|1 dt)1/2 ( if n = 2). ≤ M sup |v(t, ·)|0 (

∫ ∥K(v)∥0,−1 = (

t

Inequality (6.46) is proved.

T

0

(6.50)

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96

1/4

3/4

Now let n = 3. Then, since ∥v∥L4 (Ω) ≤ M |v|0 |v|1 |K(v)|−1 ≤ M |

n ∑

for n = 3, one has 1/2

3/2

vi v|0 ≤ M ∥v∥2L4 (Ω) ≤ M |v|0 |v|1 .

(6.51)

i=1

From inequality (6.51) it follows that ∫ T ∫ T 4/3 2/3 2 3/4 ∥K(v)∥L4/3 (0,T ;V ∗ ) = ( |K(v)|−1 dt) ≤( |v(t, ·)|0 |v(t, ·)|1 dt)3/4 0 0 ∫ T 1/2 2 ≤ M sup |v(t, ·)|0 ( |v(t, ·)|1 dt)3/4 ( if n = 3). t

0

Inequality (6.47) is proved. This completes the proof of Proposition 6.6.

□

Proposition 6.7. Function v satisfies Eq. (3.11). Proof of Proposition 6.7. Use the statement of Proposition 6.5. From Lemma 6.4 it follows that v satisfies for a.e. t the identity (2.4). We rewrite it in the form d(v, φ)/dt = ⟨ˆ g , φ⟩,

(6.52)

gˆ = f − A(v) + K(v) − C(v, z).

(6.53)

where

Find out summability properties of the summands in the right hand side of (6.53). From inequality (6.44) and estimate (6.4) it follows that ∥A(v)∥0,−1 ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 ).

(6.54)

From inequality (6.49) and estimate (6.4) it follows that ∫ t ∥C(v, z)∥0,−1 ≤ M ∥ exp((−t)/λ)|v(s, ·)|1 ds∥0 ≤ 0

M (1 + ∥f ∥0,−1 + |v 0 |0 ).

(6.55)

From estimates (6.4), (6.46) (6.47) it follows that ∥K(v)∥0,1 ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 (for n = 2).

(6.56)

∥K(v)∥L4/3 (0,T ;V ∗ ) ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 (for n = 3).

(6.57)

From (6.53) and inequalities (6.54), (6.55), (6.56) it follows that gˆ ∈ L2 (0, T ; V ∗ ) for n = 2 and the following inequality ∥ˆ g ∥0,−1 ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 holds. Let n = 3. From (6.53) and inequalities (6.54), (6.55), (6.57) for an arbitrary natural number k there follows the validity of inequality ∥ˆ g ∥L4/3 (0,T ;V ∗ ) ≤ ∥f ∥L4/3 (0,T ;V ∗ ) + ∥K(v)∥L4/3 (0,T ;V ∗ ) + ∥C(v, z)∥L4/3 (0,T ;V ∗ ) + ∥A(v)∥L4/3 (0,T ;V ∗ ) ≤ M (∥f ∥0,−1 + ∥C(v, z)∥0,−1 + ∥A(v)∥0,−1 ) + ∥K(v)∥L4/3 (0,T ;V ∗ ) ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 .

(6.58)

V.G. Zvyagin, V.P. Orlov / Nonlinear Analysis 172 (2018) 73–98

97

By Proposition 6.5, from the summability of function gˆ there follows the existence of dv/dt as a function with values in V ∗ , the equality dv/dt = gˆ for a.e. t ∈ (0, T ] and, in virtue of (6.52) and (6.53), validity of Eq. (6.43). Proposition 6.7 is established. □ From Eq. (3.11) and estimates of the terms of the function gˆ it follows the validity of inequalities ∥dv/dt∥0,1 ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 ,

for n = 2,

∥dv/dt∥L4/3 (0,T ;V ∗ ) ≤ M (1 + ∥f ∥0,−1 + |v 0 |0 )2 , k ∈ (0, T ), for n = 3. The last inequalities and (6.4) means that v ∈ W0 . Lemma 6.5 is proved. 6.5. Conclusion of the proof of Theorem 2.3. Lemmas 6.4 and 6.5 imply the assertion of Theorem 2.3. Acknowledgment The research of the first author was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The research of the second author was supported by the Russian Science Foundation (project no. 16-11-10125, Theorem 3.1). References

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