Perturbation of Stochastic Boussinesq Equations with Multiplicative White Noise

Abstract

This paper studies the Boussinesq equations perturbed by multiplicative white noise and shows the existence and uniqueness of the global solution. It also gets some regularity results for the unique solution.

1. Introduction

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation. The deterministic case has been studied systematically by many authors (e.g., see [1–3]). However, in many practical circumstances, small irregularity has to be taken into account. Thus, it is necessary to add to the equation a random force, which is in general a space-time white noise, as considered recently by many authors for other equations (see [4–11]). The random attractors of boussinesq equations with multiplicative noise have been investigated by [12]. In this paper, We will study the perturbation of stochastic boussinesq equations with multiplicative white noise.

We will consider the following stochastic two-dimension a Boussinesq equations perturbed by a multiplicative white noise of Stratonovich form: 𝑑𝑣+(𝑣⋅∇)𝑣−𝑣Δ𝑣+∇𝑝𝑑𝑡=𝑒₂𝑇−𝑇₁(𝑡),)]𝑑𝑡+𝜎𝑣∘𝑑𝑤𝑑𝑇+[(𝑣⋅∇𝑇−𝜆Δ𝑇𝑑𝑡=0,div𝑣=0. (1)

The domain occupied by the fluid is 𝐷=(0,1)×(0,1), and 𝑒₁, 𝑒₂ is the canonical basis of ℝ². The unknown 𝑣=(𝑣₁,𝑣₂), 𝑇, and 𝑝 stand for the velocity vector, temperature, and pressure, respectively. 𝑇₁ is the temperature at the top, 𝑥₂=1, while 𝑇₀=𝑇₁+1 is the temperature at the boundary below, 𝑥₂=0. The constant numbers 𝜆>0, 𝛽>0, and  𝜎>0 are related to the usual Prandtl, Grashof, and Rayleigh numbers.

𝑊(𝑡) is two-sided Wiener processes on the probability space (Ω,𝔽,𝑃), where Ω={𝜔∈𝐶(ℝ,ℝ)∶𝜔(0)=0}, 𝔽 is the Borel sigma-algebra induced by the compact-open topology of Ω, and 𝑃 is a Wiener measure.

We supplement (1) with the following boundary condition: 𝑣=0at𝑥₂=0,𝑥₂=1,𝑇=𝑇₀at𝑥₂=0,𝑇=𝑇₁=𝑇₀−1at𝑥₂=1,𝜓∣_𝑥1=0=𝜓∣_𝑥1=1for𝜓=𝑣,𝑇,𝑝,𝜕𝑣𝜕𝑥₁,𝜕𝑇𝜕𝑥₁. (2)

When an initial-valued problem is considered, we supplement these equations with 𝑣(𝑥,0)=𝑣₀(𝑥),𝑇(𝑥,0)=𝑇₀(𝑥)for𝑥∈𝐷. (3)

The existence of a compact random attractor and its Hausdorff, fractal dimension estimates have been investigated by [12]. We will solve pathwise (1)–(3). By using the Faedo-Galerkin approximation and a priori estimates, we prove the existence and uniqueness of the global solution and show that the solution continuously depends on the initial value. We also get some regularity results of the solutions.

2. Mathematical Setting and Basic Estimates

Let 𝜂∶=𝑇−𝑇₀+𝑥₂, (4) and change 𝑝 to 𝑝−𝑥₂+𝑥²₂/2; then (1) can be rewritten as 𝑑𝑣+(𝑣⋅∇)𝑣−𝛽Δ𝑣+∇𝑝𝑑𝑡=𝑒₂)𝜂𝜂𝑑𝑡+𝜎𝑣∘𝑑𝑊,𝑑𝜂+(𝑣⋅∇𝑑𝑡=𝑣₂𝑑𝑡,div𝑣=0. (5) Let the process be 𝛼(𝑡)∶=𝑒^{−𝜎𝑊(𝑡)}. (6) Then 𝑑𝛼=−𝜎𝛼∘𝑑𝑊, and if we let 𝜉∶=𝛼𝑣, (7) we get the new equations (no stochastic differential appears here) 𝑑𝜉𝑑𝑡+𝛼⁻¹(𝜉⋅∇)𝜉−𝛽Δ𝜉+𝛼∇𝑝=𝛼𝑒₂𝜂,𝑑𝜂𝑑𝑡+𝛼⁻¹(𝜉⋅∇)𝜂−𝜆Δ𝜂=𝛼⁻¹𝜉₂,div𝜉=0, (8)(9)(10) with the boundary conditions 𝜉=0at𝑥₂=0,𝑥₂=1,𝜂=0at𝑥₂=0,𝑥₂𝜓=1,_𝑥1=0=𝜓_𝑥1=1for𝜓=𝜉,𝜂,𝑝,𝜕𝜉𝜕𝑥₁,𝜕𝜂𝜕𝑥₁ (11) and the initial value conditions 𝜉(0)=𝜉₀,𝜂(0)=𝜂₀. (12)

To solve (8)–(12), we consider the Hilbert space 𝐻=𝐻₁×𝐻₂ with the scalar products (⋅,⋅) and norms |⋅|, where𝐻₂=𝐿²(𝐷) and 𝐻₁=𝜉∈𝐿²(𝐷)∶div𝜉=0,𝜉_{𝑖𝑥𝑖=0}=𝜉_{𝑖𝑥𝑖=1},𝑖=1,2. (13) We also consider the subspace 𝑉=𝑉₁×𝑉₂ of 𝐻, where 𝑉₂ is the space of functions in 𝐻¹(𝐷) vanishing at 𝑥₂=0 and 𝑥₂=1 and periodic in the direction of 𝑥₁.𝑉₂ is a Hilbert space for the scalar product and the norm 𝜂₁,𝜂₂=_𝐷grad𝜂₁grad𝜂₂𝑑𝑥,𝜂=𝜂,𝜂^1/2, (14) and 𝑉₁={𝜉∈𝑉²₂∶div𝜉=0}. We also denote by ((⋅,⋅)) and ‖⋅‖ the canonical scalar product and norm in 𝑉₁ and 𝑉.

The bilinear form 𝜇𝑢₁,𝑢₂=𝛽𝜉₁,𝜉₂+𝜆𝜂₁,𝜂₂,∀𝜉_𝑖,𝜂_𝑖∈𝑉,𝑖=1,2, (15) determines a linear isomorphism 𝐴 from 𝐷(𝐴) into 𝐻 and from 𝑉 into the dual　space 𝑉^′, defined by 𝐴𝑢₁,𝑢₂=𝜇𝑢₁,𝑢₂,∀𝑢_𝑖=𝜉_𝑖,𝜂_𝑖∈𝑉,𝑖=1,2, (16) with 𝐷(𝐴)=𝐷(𝐴₁)×𝐷(𝐴₂),  where 𝐷𝐴₁=𝜉∈𝑉₁∩𝐻²(𝐷)²∶𝜕𝜉𝜕𝑥_1𝑥1=0=𝜕𝜉𝜕𝑥_1𝑥1=1𝐷𝐴₂=𝜂∈𝑉₂∩𝐻²(𝐷)∶𝜕𝜂𝜕𝑥_1𝑥1=0=𝜕𝜂𝜕𝑥_1𝑥1=1. (17) Four spaces 𝐷(𝐴), 𝑉, 𝐻, and 𝑉^′ satisfy 𝐷(𝐴)⊂𝑉⊂𝐻⊂𝑉^′ (18) and all embedding injections are densely continuous. It is well known that 𝐴∶𝐷(𝐴)→𝐻 is self-adjoint and positive and 𝐴⁻¹ is a compact self-adjoint in 𝐻.

We also consider the trilinear forms 𝛾 on 𝑉 defined by 𝛾𝑢₁,𝑢₂,𝑢₃=𝜉₁⋅∇𝜉₂,𝜉₃+𝜉₁⋅∇𝜂₂,𝜂₃,∀𝑢_𝑖=𝜉_𝑖,𝜂_𝑖∈𝑉,𝑖=1,2,3. (19) The trilinear form 𝛾 is continuous on 𝑉 or even on𝐻¹(𝐷)²×𝐻¹(𝐷). We associate with the form 𝛾 the bilinear continuous operator 𝐵 which map 𝑉×𝑉 into 𝑉^′ and 𝐷(𝐴)×𝐷(𝐴) into 𝐻, defined by 𝐵𝑢₁,𝑢₂,𝑢₃=𝛾𝑢₁,𝑢₂,𝑢₃,∀𝑢_𝑖=𝜉_𝑖,𝜂_𝑖∈𝑉,𝑖=1,2,3. (20) Finally, we define the continuous operators 𝑅(𝑡) in 𝐻𝑅(𝑡)∶𝑢=𝜉,𝜂⟶𝑅𝑢=𝛼(𝑡)𝑒₂𝜂,𝛼⁻¹(𝑡)𝜉₂. (21) Now, we can set (8) in the operator form. If𝑢={𝜉,𝜂} is the solution of (8) and 𝜓={𝑓,𝑔} is a test function in 𝑉, we multiply (8) by 𝑓 and (9) by 𝑔, integrate over 𝐷, and add the resulting equation. The pressure term disappears and after simplification we find 𝑑𝑑𝑡𝑢,𝜓+𝛼⁻¹𝑢,𝑢,𝜓+𝜇𝑢,𝜓+𝑅(𝑡)𝑢,𝜓=0,∀𝜓∈𝑉, (22) which can be reinterpreted as 𝑑𝑢𝑑𝑡+𝐴𝑢+𝛼⁻¹(𝑡)𝐵(𝑢,𝑢)+𝑅(𝑡)𝑢=0. (23) Note that this equation differs from the determined case, and in determined case, the family 𝑅(𝑡) of operator is independent of the time 𝑡. Initial condition (12) can be reinterpreted as 𝑢(0)=𝑢₀∶=𝜉₀,𝜂₀. (24) To solve (23)-(24), we also need some Sobolev norm estimates on the bilinear 𝐵 and the operators 𝑅 and 𝐴.

Lemma 1. The bilinear operators 𝐵∶𝑉×𝑉→𝑉^′ and 𝐷(𝐴)×𝐷(𝐴)→𝐻 are continuous and satisfy(i)(𝐵(𝑢,𝑣),𝑣)=0,  for all  𝑢,𝑣∈𝑉,(ii)|(𝐵(𝑢,𝑣),𝑤)|≤𝑐₁|𝑢|^𝜃1‖𝑢‖^1−𝜃1‖𝑣‖‖𝑤‖^𝜃1|𝑤|^1−𝜃1,  for all  𝑢,𝑣,𝑤∈𝑉,(iii)|𝐵(𝑢,𝑣)|+|𝐵(𝑣,𝑢)|≤𝑐₂‖𝑢‖‖𝑣‖^1−𝜃2|𝐴𝑣|^𝜃2,  for all  𝑢∈𝑉, 𝑣∈𝐷(𝐴),(iv)|𝐵(𝑢,𝑣)|≤𝑐₃|𝑢|^𝜃3‖𝑢‖^1−𝜃3|𝐴𝑣|^𝜃3‖𝑣‖^1−𝜃3,  for all  𝑢∈𝑉, 𝑣∈𝐷(𝐴),

where  𝑐₁, 𝑐₂, 𝑐₃  are appropriate constants and  𝜃_𝑖∈[0,1),  𝑖=1,2,3.

Proof. The proof is the same as the deterministic case (see [10]).

Lemma 2. The linear continuous operators 𝑅∶𝑉→𝑉^′ and 𝐷(𝐴)→𝐻 satisfy |𝑅(𝑡)𝑢|≤𝑐₄(𝛼(𝑡))+𝛼⁻¹(𝑡𝑢‖𝑅(𝑡))‖,∀𝑢∈𝑉,∀𝑡≥0,|(𝑢,𝑢)|≤𝑐₄𝛼(𝑡)+𝛼⁻¹(𝑡)‖𝑢‖|𝑢|,∀𝑢∈𝑉,∀𝑡≥0. (25)(26)

Proof. By (21), we have |𝑅𝑢|=𝛼𝑒₂𝜂+𝛼⁻¹𝜉₂≤𝛼𝜂+𝛼⁻¹𝜉≤𝛼+𝛼⁻¹𝜉+𝜂 (27) which implies by the Poincare inequality |𝑢|≤𝑐₄‖𝑢‖,for𝑢∈𝑉 (28) that (25) holds true. Since |(𝑅𝑢,𝑢)|≤|𝑅𝑢||𝑢|, it follows from (25) that (26) holds true.

Lemma 3. The bilinear form 𝜇 on 𝑉×𝑉 satisfies 𝑐₅‖𝑢‖²≤𝜇(𝑢,𝑢)≤𝑐₆‖𝑢‖²,𝑓𝑜𝑟𝑢∈𝑉. (29)

Proof. By (15), we have 𝜇(𝑢,𝑢)=𝛽𝜉²+𝜆𝜂²≤𝛽+𝜆𝜉+𝜂²=𝛽+𝜆‖𝑢‖²,𝜇(𝑢,𝑢)≥min𝛽,𝜆𝜉²+𝜂²≥12min𝛽,𝜆𝜉²+𝜂², (30) which imply (28).

3. Existence and Uniqueness

In this section, we will prove the existence and uniqueness of the global solution of (23)-(24), equivalently (8)–(12) or (1)–(3). We are working almost surely for 𝜔∈Ω.

Theorem 4. Assume that 𝑢₀∈𝐻, then there exists a unique solution of (23)-(24), such that 𝑢∈𝐶([0,∞),𝐻)∩𝐿²_loc(0,∞;𝑉), (31) and the mapping 𝑢₀↦𝑢(𝑡) is continuous from H into D(A), for all 𝑡>0.

Proof. Since 𝐴⁻¹∶𝐻→𝐷(𝐴) is a self-adjoint compact operator in 𝐻, it follows from a classical spectral theorem that there exists a sequence𝜆_𝑗∶0<𝜆₁≤𝜆₂≤⋯,𝜆_𝑗→∞ and a family of elements 𝑤_𝑗∈𝐷(𝐴) which is completely orthogonal in 𝐻 such that 𝐴𝑤_𝑗=𝜆_𝑗𝑤_𝑗,∀𝑗. (32) For each 𝑚, we look for an approximate solution 𝑢_𝑚 of the following form: 𝑢_𝑚(𝑡)=^𝑚_𝑖=1𝑔_𝑖𝑚(𝑡)𝑤_𝑖 (33) satisfying 𝑑𝑢𝑑𝑡^𝑚,𝑤_𝑗+𝜇𝑢_𝑚,𝑤_𝑗+𝛼⁻¹𝛾𝑢_𝑚,𝑢_𝑚,𝑤_𝑗+𝑅𝑢_𝑚,𝑤_𝑗=0,𝑗=1,2,…,𝑚 (34) and initial condition 𝑢_𝑚(0)=𝑃_𝑚𝑢₀, (35) where 𝑃_𝑚 is the projector in 𝐻 (or 𝑉) on the space spanned by𝑤₁,𝑤₂,…,𝑤_𝑚. Since 𝐴 and 𝑃_𝑚 commute, the above equation is also equivalent to 𝑑𝑢_𝑚𝑑𝑡+𝐴𝑢_𝑚+𝛼⁻¹𝑃_𝑚𝐵𝑢_𝑚,𝑢_𝑚+𝑃_𝑚𝑅𝑢_𝑚=0, (36) where 𝑃_𝑚𝑅𝑢_𝑚=^𝑚_𝑗=1𝑔_𝑗𝑚(𝑡)𝑃_𝑚𝑅𝑤_𝑗, (37) in view of the linearity of 𝑃_𝑚,𝑅.
The existence of 𝑢_𝑚 on any finite interval [0,𝑇_𝑚) follows from standard results of the existence of solutions of ordinary differential equations that 𝑇_𝑚=+∞ is a consequence of these results and of the following priori estimates: 𝑢_𝑚remainsboundedin𝐿^∞(0,𝑇;𝐻)∩𝐿²(0,𝑇;𝑉),∀𝑇>0. (38) Indeed, multiplying (34) by 𝑔_𝑗𝑚, summing these relations for 𝑗=1,2,…,𝑚, and noting that 𝛼⁻¹𝜇(𝑢_𝑚,𝑢_𝑚,𝑢_𝑚)=0 (by Lemma 1), we find 12𝑑𝑢𝑑𝑡^𝑚2+𝜇𝑢_𝑚,𝑢_𝑚+𝑅𝑢_𝑚,𝑢_𝑚=0, (39) which implies by Lemma 2, (29), and the Young inequality that 12𝑑𝑢𝑑𝑡^𝑚2+𝑐₅𝑢_𝑚²≤𝑅𝑢_𝑚,𝑢_𝑚≤𝑐₄sup_{0≤𝑡≤𝑇}𝛼(𝑡)+𝛼⁻¹(𝑡)𝑢_𝑚⋅𝑢_𝑚≤𝑐₅2𝑢_𝑚²+𝑐^′₄𝑢_𝑚², (40) that is, 𝑑𝑢𝑑𝑡^𝑚2+𝑐₅𝑢_𝑚²≤𝑐^′𝑢_𝑚², (41) where 𝑐₅ is defined in (29) and 𝑐^′ is a appropriate constant. Using the classical Gronwall lemma we find 𝑢_𝑚(𝑡)²≤𝑢_𝑚(0)²𝑒^𝑐′𝑇≤𝑢₀²𝑒^𝑐′𝑇=𝑀₁,∀0<𝑡≤𝑇. (42) Integrating (41) for 𝑡 from 0 to 𝑇 and using above estimates we have ^𝑇₀𝑢_𝑚²≤1𝑑𝑡𝑐₅𝑢_𝑚(0)²−𝑢_𝑚(𝑇)²+1𝑐₅^𝑇₀𝑢_𝑚(𝑡)²𝑑𝑡≤𝑀₂, (43) where 𝑀₁and 𝑀₂ are independent of 𝑚. Thus, we have proved (38).
We also claim that 𝑑𝑢_𝑚𝑑𝑡remainsboundedin𝐿²0,𝑇;𝑉^′. (44) Indeed, it follows from Lemma 1 that |𝐵(𝑢_𝑚,𝑢_𝑚)|𝑉^′≤𝑐|𝑢_𝑚|‖𝑢_𝑚‖ with appropriate constant c, which, together with (38), implies that 𝐵(𝑢_𝑚,𝑢_𝑚) and thus 𝑃_𝑚𝐵(𝑢_𝑚,𝑢_𝑚) remain bounded in 𝐿²(0,𝑇;𝑉^′). Since both operators 𝑅∶𝑉→𝑉^′ and 𝐴∶𝑉→𝑉^′ are continuous (Lemmas 2 and 3), it follows from (38) that 𝐴𝑢_𝑚,𝑅𝑢_𝑚 and thus 𝑃_𝑚𝑅𝑢_𝑚 remain bounded in 𝐿²(0,𝑇;𝑉^′). Therefore, by (36), 𝑑𝑢_𝑚𝑑𝑡=−𝐴𝑢_𝑚−𝛼⁻¹𝑃_𝑚𝐵𝑢_𝑚,𝑢_𝑚−𝑃_𝑚𝑅𝑢_𝑚 (45) remains bounded in 𝐿²(0,𝑇;𝑉^′), which proved (44).
By weak compactness, it follows from (38) and (44) that there exists a 𝑢∈𝐿^∞(0,𝑇;𝐻)∩𝐿²(0,𝑇;𝑉), for all 𝑇>0 subsequence still denoted by 𝑢_𝑚, such that 𝑢_𝑚⟶𝑢in𝐿²()𝑢0,𝑇;𝑉weakly,_𝑚⟶𝑢in𝐿^∞(0,𝑇;𝐻)weaklystar,𝑑𝑢_𝑚⟶𝑑𝑡𝑑𝑢𝑑𝑡in𝐿²0,𝑇;𝑉^′weakly. (46) We pass to the limit in (34) and find that 𝑑𝑑𝑡𝑢,𝜑+𝜇𝑢,𝜑+𝛼⁻¹𝛾𝑢,𝑢,𝜑+𝑅𝑢,𝜑,∀𝜑∈𝑉, (47) which implies that 𝑢 satisfies (23). In particular,  𝑢^′=(𝑑𝑢/𝑑𝑡)∈𝐿²(0,𝑇;𝑉^′)∈𝐿¹(0,𝑇;𝑉^′). This implies by [10, Lemma II.3.1] that 𝑢 is almost everywhere equal to a continuous function from [0,𝑇] into 𝑉^′. Therefore initial condition (24) follows by a passage to the limit in (35). 𝑢∈𝐶([0,𝑇],𝐻) follows from [10, Lemma II.3.2] and the facts that 𝐻⊂𝑉⊂𝑉^′ and 𝑢^′∈𝐿²(0,𝑇;𝑉^′). Furthermore, if we show that uniqueness, then the fact that 𝑢∈𝐶([0,𝑇],𝐻), for all 𝑇>0, implies that 𝑢∈𝐶([0,∞),𝐻).
To prove the uniqueness and continuous dependence of 𝑢(𝑡) on 𝑢₀ (in 𝐻), we let 𝑢 be a solution of (23)-(24) such that 𝑢∈𝐶([0,∞),𝐻)∩𝐿²_loc(0,∞;𝑉). Similar to (39), 𝑢  must satisfy the energy equality 12𝑑|𝑢|𝑑𝑡²+𝜇(𝑢,𝑢)+(𝑅𝑢,𝑢)=0. (48) By using Lemmas 1–3 and Gronwall lemma, we get the following similar estimates: |𝑢(𝑡)|²≤|𝑢(0)|²𝑒^𝑐𝑡, (49) which has proved the continuous dependence. For the uniqueness, we let 𝑢₁,𝑢₂ be two solutions of (23)-(24) and 𝑢=𝑢₁−𝑢₂. Then 𝑢 is also a solution with 𝑢(0)=0. Thus, (49) implies that |𝑢(𝑡)|²≤0, that is, 𝑢₁=𝑢₂.

4. Regularity Results

In this section, we will consider further regularity results for the unique solution. The main result is that 𝑢∈𝐷(𝐴), and thus 𝑢∈𝐻²(𝐷)²×𝐻²(𝐷) provided the initial function 𝑢₀∈𝑉. More precisely, we have the following.

Theorem 5. Assume that 𝑢₀∈𝑉, and let 𝑢 be the unique solution of (23)-(24). Then, 𝑢∈𝐶([0,∞),𝑉)∩𝐿²_loc(0,∞;𝐷(𝐴)). (50)

Proof. Let 𝑢_𝑚 be the approximate solution (33) in the proof of Theorem 4. We first claim that 𝑢_𝑚remainsboundedin𝐿^∞(0,𝑇;𝑉)∩𝐿²(0,𝑇;𝐷(𝐴)),∀𝑇>0. (51) Indeed, multiplying (34) by 𝜆_𝑗𝑔_𝑗𝑚, summing these relations for 𝑗=1,2,…,𝑚, and using (32), we find 𝑑𝑢_𝑚𝑑𝑡,𝐴𝑢_𝑚+𝜇𝑢_𝑚,𝐴𝑢_𝑚+𝛼⁻¹𝛾𝑢_𝑚,𝑢_𝑚,𝐴𝑢_𝑚+𝑅𝑢_𝑚,𝐴𝑢_𝑚=0. (52) By Lemma 1(iv) and the Young inequality, we find 𝛼⁻¹𝛾𝑢_𝑚,𝑢_𝑚,𝐴𝑢_𝑚=𝛼⁻¹𝐵𝑢_𝑚,𝑢_𝑚,𝐴𝑢_𝑚≤sup_{0≤𝑠≤𝑇}𝛼⁻¹(𝑠)⋅𝑐₃𝑢_𝑚^𝜃3𝑢_𝑚^2(1−𝜃3)𝐴𝑢_𝑚^(1+𝜃3)≤14𝐴𝑢_𝑚²+𝑐^′₃(𝑇)𝑢_𝑚⁴|𝑢|^{2𝜃3/(1−𝜃3)}. (53) For 0≤𝑡≤𝑇, by Lemma 2, (25), and the Young inequality, we have 𝑅𝑢_𝑚,𝐴𝑢_𝑚≤sup_{0≤𝑠≤𝑇}𝛼(𝑠)+𝛼⁻¹(𝑠)𝑐₄𝑢_𝑚𝐴𝑢_𝑚≤14𝐴𝑢_𝑚²+𝑐^′₄𝑢_𝑚². (54) Noting also that 𝑑𝑢_𝑚𝑑𝑡,𝐴𝑢_𝑚=𝐴𝑢_𝑚,𝑢^′_𝑚=𝜇𝑢_𝑚,𝑢^′_𝑚=12𝑑𝜇𝑢𝑑𝑡^𝑚,𝑢_𝑚 (55) and 𝜇(𝑢_𝑚,𝐴𝑢_𝑚)=|𝐴𝑢_𝑚|², we find from (52) and all the above estimates that 𝑑𝜇𝑢𝑑𝑡^𝑚,𝑢_𝑚+𝐴𝑢_𝑚²≤2𝑐^′₄𝑢_𝑚²+2𝑐^′₃𝑢_𝑚⁴𝑢_𝑚^{2𝜃3/(1−𝜃3)}. (56) By (38), 𝑢_𝑚 is bounded in 𝐿^∞(0,𝑇;𝐻). This, together with Lemma 3, implies that (56) can be rewritten as 𝑑𝜇𝑢𝑑𝑡^𝑚,𝑢_𝑚+𝐴𝑢_𝑚²≤2𝑐1+𝑢_𝑚²𝜇𝑢_𝑚,𝑢_𝑚,0≤𝑡≤𝑇 (57) for some appropriate constant 𝑐>0. By Gronwall lemma, it follows from (57) and (38) that 𝜇𝑢_𝑚(𝑡),𝑢_𝑚(𝑡)≤𝑢_𝑚(0)²exp^𝑡₀𝑐1+𝑢_𝑚(𝑠)²𝑑𝑠≤𝑀₁(𝑇),0≤𝑡≤𝑇, (58) which implies by Lemma 3 again that 𝑢_𝑚 remains bounded in 𝐿^∞(0,𝑇;𝑉). Integrating in (57) from 𝑡=0 to  𝑡=𝑇, we have ^𝑇₀𝐴𝑢_𝑚(𝑡)²𝑑𝑠≤2𝑀₁+^𝑇₀𝑀₁1+𝑢_𝑚(𝑡)²𝑑𝑠≤𝑀₂. (59) which proved the second argument of (51), and thus (51) holds.
Taking the limit in (51) (by weak compactness), we then find that 𝑢 is in 𝐿^∞(0,𝑇;𝑉)∩𝐿²(0,𝑇;𝐷(𝐴)). We need also to prove that u is continuous from [0,𝑇] into 𝑉. This is proved as follows.
Since 𝑢∈𝐶([0,𝑇],𝐻)∩𝐿^∞(0,𝑇;𝑉) and 𝑉⊂𝐻 with densely continuous injection, it follows from [10, Lemma II.3.3] that 𝑢∶[0,𝑇]→𝑉 is weakly continuous; that is, 𝑡↦((𝑢(𝑡),𝑣))  is continuous for every 𝑣∈𝑉. Similarly 𝑡↦𝜇((𝑢(𝑡),𝑣))  is continuous for every 𝑣∈𝑉. Thus, by taking the limit in (52) and applying [10, Lemma II.3.2], we obtain an equality similar to (52) for 𝑢: 𝑑𝜇()||𝑑𝑡𝑢,𝑢+2𝐴𝑢²+2𝛼⁻¹𝜆(𝑢,𝑢,𝐴𝑢)+2(𝑅𝑢,𝑢)=0, (60) which holds in the distribution sense on (0,𝑇). Since 𝑢∈𝐿^∞(0,𝑇;𝑉)∩𝐿²(0,𝑇;𝐷(𝐴)),  it follows from Lemma 1 and Lemma 2 that |𝐴𝑢|²+𝛼⁻¹𝛾(𝑢,𝑢,𝐴𝑢)+(𝑅𝑢,𝑢)∈𝐿¹(0,𝑇;ℝ) (61) and thus (𝑑/𝑑𝑡)𝜇(𝑢(𝑡),𝑢(𝑡))∈𝐿¹(0,𝑇;ℝ), which implies by [10, Lemma II.3.1] that the function 𝑡↦𝜇(𝑢(𝑡),𝑢(𝑡)) is continuous. Therefore, since 𝜇(𝜑,𝜑)^1/2 is a norm on 𝑉 equivalent to ‖𝜑‖ (by Lemma 3), it follows that 𝑢∶[0,𝑇]→𝑉 is continuous for the norm topology.

Acknowledgment

This work was supported by the Foundation of Science and Technology Project of Chongqing Education Commission (KJ100513).