Global existence of solutions for hyperbolic Navier–Stokes equations in three space dimensions

Abstract

We consider a hyperbolic quasilinear perturbation of the Navier–Stokes equations in three space dimensions. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works. Furthermore, we prove the convergence of solutions to relaxed system towards solutions to the classical Navier–Stokes problem.

Keywords

Hyperbolic Navier–Stokes equations global existence quasilinear hyperbolic equations

1. Introduction

The Navier–Stokes equations describe the time evolution of solutions of mathematical models of viscous incompressible fluids. $\begin{matrix} (1.1) & (NS) \{\begin{matrix} v_{t} - ν Δ v + (v . \nabla) v = - \nabla p + f, \\ div v = 0, \\ v (0, y) = v_{0}, \end{matrix} \end{matrix}$ with $ν > 0$ being the viscosity, for the velocity vector $v = v (t, y) : (0, \infty) \times R^{3} \to R^{3}$ of a fluid, $p = p (t, y) : (0, \infty) \times R^{3} \to R$ the pressure and f the forcing term.

As is well know, the existence of a weak solution is guaranteed under a minimal assumption on the initial data in every space dimension. The uniqueness of weak solution and global existence of strong solution is known in two space dimensions (for more details see Leray [10] and Hopf [9]). But the strong global existence and uniqueness in three space dimensions remain open. Numerous authors proved global existence and uniqueness of strong solution but under very restrictive additional regularity and smallness assumptions on the initial data, see for example [5–7,18].

Thus, a natural question arises, if we consider a hyperbolic perturbation of the Navier–Stokes equations, can we prove the global existence for larger initial data, which approaches, in some sense, the classical Navier–Stokes equations?

This problem has previously been studied by several authors. A first hyperbolic perturbation of (NS) has been obtained after relaxation of the Euler equations and rescaling variables $\begin{matrix} (1.2) & \{\begin{matrix} ε u_{t t}^{ε} - ν Δ u^{ε} + u_{t}^{ε} + (u^{ε} . \nabla) u^{ε} = - \nabla p + f^{ε}, \\ div u^{ε} = 0, \\ (u^{ε}, u_{t}^{ε}) (0, y) = (u_{0}^{ε}, u_{1}^{ε}) . \end{matrix} \end{matrix}$

In [1] Brenier, Natalini and Puel proved the global existence and uniqueness in a two-dimensional space under smallness condition of the initial data in $H^{2} {(R^{2})}^{2} \times H^{1} {(R^{2})}^{2}$ and without a forcing term. Thanks to a Strichartz estimate, Paicu and Raugel in [12] improved this result. They stated the global existence and uniqueness under smallness condition of the initial data in $H^{1} {(R^{2})}^{2} \times L^{2} {(R^{2})}^{2}$ . In [11], the authors proved global existence and uniqueness in three-dimensional space under a smallness condition of the initial data in $H^{1 + δ} {(R^{3})}^{3} \times H^{δ} {(R^{3})}^{3}$ , for $δ > 0$ . In [8], Hachicha used a modulated energy method to improve the results of [1,11,12]. She relaxed the regularity of the initial data in two and three space dimensions. We recall that, Cattaneo [4] introduced this perturbation as a perturbation of the heat equations.

In this paper, we consider another hyperbolic perturbation of (NS), which is obtained by replacing a Fourier type law by the law of Cattaneo $\begin{matrix} (1.3) & {(HNS)}_{ε} \{\begin{matrix} ε u_{t t}^{ε} - ν Δ u^{ε} + u_{t}^{ε} + ε (u^{ε} . \nabla) u_{t}^{ε} + ((ε u_{t}^{ε} + u) . \nabla) u^{ε} = - \nabla p + f^{ε}, \\ div u^{ε} = 0, \\ (u^{ε}, u_{t}^{ε}) (0, y) = (u_{0}^{ε}, u_{1}^{ε}) . \end{matrix} \end{matrix}$ We note that this hyperbolic perturbation differs from the above one. Due to the quadratic nonlinearities $ε (u^{ε} . \nabla) u_{t}^{ε}$ , the system ${(HNS)}_{ε}$ is quasilinear. However, the system (1.2) is still semilinear, therefore the proofs of the classical Navier–Stokes equations remain applicable. Also, the Cattaneo type law provides a system exponentially unstable, in contrast with the Fourier type law, which further complicates the proofs.

For $f^{ε} = 0$ , this relaxation has been treated first by [2] and [3]. Racke and Saal proved in [14] the well-posedness under a smallness condition on the initial data in $H^{m + 2} {(R^{n})}^{n} \times H^{m + 1} {(R^{n})}^{n}$ , for $m > \frac{n}{2}$ integer. Using Klainerman and Ponce method they proved in [15] the global existence under a smallness condition on the initial data in $H^{m + 2} {(R^{n})}^{n} \times H^{m + 1} {(R^{n})}^{n}$ , for $m ⩾ 12$ integer. Schöwe [16,17] improves this result. For the well-posedness, he relaxes the dependence of the existence time on the initial data and proves the global existence in three-dimensional space under a smallness condition of the initial data in $H^{m + 2} {(R^{3})}^{3} \times H^{m + 1} {(R^{3})}^{3}$ , for $m ⩾ 4$ integer. Moreover, Schöwe proves the convergence of solution to the hyperbolic perturbed problem ${(HNS)}_{ε}$ towards solution to the incompressible Navier–Stokes (NS).

The aim of this paper is to improve the results of [15–17]. First, we improve the global existence results in three directions, by adding a forcing term, requiring less regularity on the initial data and allowing a larger size of the initial data and forcing term. Second, we prove the convergence of solution to ${(HNS)}_{ε}$ towards solution to (NS) under weaker assumptions on the initial data.

We begin our study by proving the global existence of ${(HNS)}_{ε}$ in $H^{4} {(R^{3})}^{3} \times H^{3} {(R^{3})}^{3}$ . More precisely, we prove the following Theorem.

Theorem 1.1.
There exist positive constants $ε_{0}$ , R such that for all $0 < ε ⩽ ε_{0}$ , if the initial data $(u_{0}^{ε}, u_{1}^{ε}) \in H^{4} {(R^{3})}^{3} \times H^{3} {(R^{3})}^{3}$ and the forcing term $f^{ε} \in L^{1} ((0, + \infty), L^{2} {(R^{3})}^{3}) \cap L^{2} ((0, + \infty), H^{3} {(R^{3})}^{3})$ satisfy $\begin{array}{l} (1.4) & \sum_{| α | ⩽ 3} ε^{| α | - \frac{1}{2}} {‖ \nabla^{α} u_{0}^{ε} ‖}_{L^{2}}^{2} + ε^{| α | + \frac{3}{2}} {‖ \nabla^{α} u_{1}^{ε} ‖}_{L^{2}}^{2} + ε^{| α | + \frac{1}{2}} {‖ \nabla^{α} \nabla u_{0}^{ε} ‖}_{L^{2}}^{2} ⩽ R, \\ (1.5) & \sum_{| α | ⩽ 3} ε^{| α | + \frac{1}{2}} {‖ \nabla^{α} f^{ε} ‖}_{L^{2} (L^{2})}^{2} + ε^{- \frac{1}{2}} {‖ f^{ε} ‖}_{L^{1} (L^{2})}^{2} ⩽ R, \end{array}$ then the System ${(HNS)}_{ε}$ admits a unique global solution $u^{ε} \in C^{1} (R^{+}, H^{3} {(R^{3})}^{3}) \cap C (R^{+}, H^{4} {(R^{3})}^{3})$ .

Once the global existence of the hyperbolic perturbed quasilinear Navier–Stokes equations is established, we prove in the next theorem, the convergence of solution to relaxed system towards solution to the classical Navier–Stokes equations.
Theorem 1.2.
Let $v_{0} \in H^{3} {(R^{3})}^{3}$ be a divergence-free vector field.

Let $(u_{0}^{ε}, u_{1}^{ε}) \in H^{4} {(R^{3})}^{3} \times H^{3} {(R^{3})}^{3}$ the initial data and $f^{ε} \in L^{1} ((0, + \infty), L^{2} {(R^{3})}^{3}) \cap L^{2} ((0, + \infty), H^{3} {(R^{3})}^{3})$ the forcing term satisfying the hypotheses of Theorem 1.1 . Assume, moreover, that there exist positive constants $ε_{1}$ , M such that for all $0 < ε ⩽ ε_{1}$ , $\begin{array}{l} (1.6) & ε^{- 1} {‖ u_{0}^{ε} - v_{0} ‖}_{L^{2}} + ε^{- \frac{1}{2}} {‖ \nabla (u_{0}^{ε} - v_{0}) ‖}_{L^{2}} + {‖ u_{1}^{ε} - v_{1} ‖}_{L^{2}} ⩽ M, \\ (1.7) & {‖ f_{t}^{ε} ‖}_{L^{2} ({\dot{H}}^{- 1})}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2} (L^{2})}^{2} ⩽ M, \end{array}$ where $v_{1} : = ν Δ v_{0} - P (v_{0} . \nabla) v_{0} + P f^{ε} (0, y)$ .

Then the global solution $u^{ε}$ of the system ${(HNS)}_{ε}$ converges, when ε goes to 0, towards the solution of the incompressible Navier–Stokes equations with $v_{0}$ as initial data, in the space $L_{loc}^{\infty} (R^{+}, H^{1} {(R^{3})}^{3})$ .

Moreover, for all positive time T, there exists a positive constant $K (T, v)$ , depending only on T and v, such that for all $0 ⩽ t ⩽ T$ $\begin{array}{rcl} {‖ u^{ε} (t) - v (t) ‖}_{L^{2}} + ε^{\frac{1}{2}} ‖ \nabla {(u^{ε} (t) - v (t) ‖}_{L^{2}} ⩽ ε K (T, v) . \end{array}$
Remark 1.3.

In Theorem 1.1, we improve the global existence results of [15–17]. The basic idea of the proof is to use a refined energy method. Interestingly, we prove the global existence under the assumption (1.4). Obviously, we relax the regularity of the initial data $(u_{0}^{ε}, u_{1}^{ε}) \in H^{4} {(R^{3})}^{3} \times H^{3} {(R^{3})}^{3}$ . Exploiting the dependence with respect to ε in (1.4), we remove the smallness assumption on the initial data since ε is small enough.

We also emphasize the importance of our result by adding a forcing term, which has not been considered in previous results.

Since our global existence theorem will be proved for less regular initial data, we have an important consequence for the relaxation limit. We improve the result of [16] in Theorem 1.2 and we prove the convergence of solution to ${(HNS)}_{ε}$ towards solution to (NS) in less regular Sobolev space $H^{1} {(R^{3})}^{3}$ . In contrast to [16], this convergence is in high Sobolev space $H^{m + 2} {(R^{3})}^{3}$ $m ⩾ 2$ integer.

This paper is organized as follows. In Section 2, we introduce notation and some preliminary results. In Section 3, we prove the global existence and uniqueness of the hyperbolic perturbed quasilinear Navier–Stokes equations. Section 4 is devoted to the study of the convergence of solution to ${(HNS)}_{ε}$ towards solution to (NS).
2. Notation and preliminary results

We start by introducing some notation. Let $L^{p} (R^{3})$ with $1 ⩽ p ⩽ \infty$ denotes the standard Lebesgue space with norm $‖ \cdot ‖_{L^{p}}$ . For the Hilbert space $L^{2} (R^{3})$ , we note $(\cdot, \cdot)$ the scalar product. Also, $H^{m} (R^{3})$ denotes the Sobolev space with the norm $\begin{matrix} ‖ u ‖_{H^{m}} = ‖ u ‖_{m} = {(\sum_{| α | ⩽ m} {‖ \nabla^{α} u ‖}_{L^{2}}^{2})}^{\frac{1}{2}}, \end{matrix}$ where $α = (α_{1}, α_{2}, α_{3}) \in N_{0}^{3}$ , $\nabla^{α} = \partial_{1}^{α_{1}} \partial_{2}^{α_{2}} \partial_{3}^{α_{3}}$ and $| α | = α_{1} + α_{2} + α_{3}$ .

$Ḣ^{m} (R^{3})$ denotes the homogeneous Sobolev space with the seminorm $\begin{matrix} ‖ u ‖_{{\dot{H}}^{m}} = | u |_{m} = {(\sum_{| α | = m} {‖ \nabla^{α} u ‖}_{L^{2}}^{2})}^{\frac{1}{2}} . \end{matrix}$ We need the following Moser-type inequalities (see [13]).

Lemma 2.1.
Let $m \in N$ . Then there is a constant $c = c (m) > 0$ such that for all $f, g \in H^{m} (R^{3}) \cap L^{\infty} (R^{3})$ and for all $α \in N_{0}^{3}$ , $| α | ⩽ m$ , the following inequality holds $\begin{matrix} {‖ \nabla^{α} (f g) ‖}_{L^{2}} ⩽ c (‖ f ‖_{L^{\infty}} {‖ \nabla^{m} g ‖}_{L^{2}} + {‖ \nabla^{m} f ‖}_{L^{2}} ‖ g ‖_{L^{\infty}}) . \end{matrix}$

Rather than studying an equation which is dependent of ε with fixed initial data, it is more convenient to fix the equation and transform the ε-dependence into the initial data. For this purpose, we introduce the diffusive scaling: $\begin{matrix} u^{ε} (τ, y) = \frac{1}{\sqrt{ε}} u (\frac{τ}{ε}, \frac{y}{\sqrt{ε}}), p^{ε} (τ, y) = \frac{1}{ε} p (\frac{τ}{ε}, \frac{y}{\sqrt{ε}}), f^{ε} (τ, y) = \frac{1}{ε \sqrt{ε}} f (\frac{τ}{ε}, \frac{y}{\sqrt{ε}}), \end{matrix}$ and $t = \frac{τ}{ε}, x = \frac{y}{\sqrt{ε}}$ .

This scaling transforms the System ${(HNS)}_{ε}$ as follows $\begin{matrix} (2.1) & (HNS) \{\begin{matrix} u_{t t} - ν Δ u + u_{t} + (u . \nabla) u_{t} + (u_{t} . \nabla) u + (u . \nabla) u = - \nabla p + f, \\ div u = 0, \\ (u, u_{t}) (0, x) = (u_{0}, u_{1}) (x) = (\sqrt{ε} u_{0}^{ε} (\sqrt{ε} x), ε^{\frac{3}{2}} u_{1}^{ε} (\sqrt{ε} x)) . \end{matrix} \end{matrix}$ We remark that for $m ⩾ 0$ , $p ⩾ 1$ , $\begin{array}{l} | u |_{m} = ε^{\frac{m}{2} - \frac{1}{4}} {| u^{ε} |}_{m}, | u_{t} |_{m} = ε^{\frac{m}{2} + \frac{3}{4}} {| u_{t}^{ε} |}_{m}, \\ ‖ u ‖_{L^{p}} = ε^{\frac{- 3}{2 p} + \frac{1}{2}} {‖ u^{ε} ‖}_{L^{p}}, \end{array}$ and $\begin{matrix} ‖ f ‖_{L^{p} ({\dot{H}}^{m})} = ε^{\frac{m}{2} + \frac{3}{4} - \frac{1}{p}} {‖ f^{ε} ‖}_{L^{p} ({\dot{H}}^{m})} . \end{matrix}$ In order to eliminate the pressure term, we apply the Leray projector $P$ which maps $L^{2} {(R^{3})}^{3}$ into $L_{σ}^{2} {(R^{3})}^{3} = {f \in L^{2} {(R^{3})}^{3}; div f = 0}$ to (HNS), we obtain the equation $\begin{matrix} (2.2) & \{\begin{matrix} u_{t t} - ν Δ u + u_{t} = - P (u . \nabla) u - P (u . \nabla) u_{t} - P (u_{t} . \nabla) u + P f, \\ div u = 0, \\ (u, u_{t}) (0, y) = (u_{0}, u_{1}) \in H^{4} {(R^{3})}^{3} \times H^{3} {(R^{3})}^{3} . \end{matrix} \end{matrix}$ Once knowing u, one can determine the pressure p.
3. Global existence in $H^{4} \times H^{3}$

In this section, we prove the global existence for (HNS) using energy estimates.

First, we recall the local existence result from Racke and Saal [14]. For each $(u_{0}, u_{1}) \in H^{m + 2} \times H^{m + 1}$ , $m > \frac{3}{2}$ integer, there exists a time $T > 0$ and a unique solution u to the equation (HNS) satisfying $u \in C^{2} ([0, T], H^{m}) \times C^{1} ([0, T], H^{m + 1}) \times C ([0, T], H^{m + 2})$ .

We introduce the energy defined by $\begin{matrix} E (t) = \sum_{| α | ⩽ 3} \frac{1}{2} {‖ \nabla^{α} (u + u_{t}) ‖}_{L^{2}}^{2} + \frac{1}{2} {‖ \nabla^{α} u_{t} ‖}_{L^{2}}^{2} + ν {‖ \nabla^{α} \nabla u ‖}_{L^{2}}^{2}, \end{matrix}$ and $\begin{matrix} Y (t) = \sum_{| α | ⩽ 3} {‖ \nabla^{α} u_{t} ‖}_{L^{2}}^{2} + ν {‖ \nabla^{α} \nabla u ‖}_{L^{2}}^{2} . \end{matrix}$ We shall prove that the energy $E (t)$ is bounded in order to show that the local solution is global. For this purpose, we prove the following Lemma.

Lemma 3.1.
There exist positive constants C and K such that for all $0 ⩽ t ⩽ T$ $\begin{array}{rcl} (3.1) & \frac{d E (t)}{d t} + (\frac{1}{2} - C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t))) Y (t) ⩽ ‖ f ‖_{L^{2}} ‖ u + u_{t} ‖_{L^{2}} + K \sum_{| α | ⩽ 3} {‖ \nabla^{α} f ‖}_{L^{2}}^{2} . \end{array}$
Proof.
Let $0 ⩽ | α | ⩽ 3$ . Applying $\nabla^{α}$ to the equation (2.2) and taking the $L^{2}$ inner product of this equation with $\nabla^{α} (u + 2 u_{t})$ , we obtain $\begin{array}{l} \frac{d}{d t} (\frac{1}{2} {‖ \nabla^{α} (u + u_{t}) ‖}_{L^{2}}^{2} + \frac{1}{2} {‖ \nabla^{α} u_{t} ‖}_{L^{2}}^{2} + ν {‖ \nabla^{α} \nabla u ‖}_{L^{2}}^{2}) + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}^{2} + ν {‖ \nabla^{α} \nabla u ‖}_{L^{2}}^{2} \\ (3.2) & = \sum_{i = 1}^{4} (\nabla^{α} N_{i}, \nabla^{α} (u + 2 u_{t})), \end{array}$ where $N_{1} = - P (u . \nabla) u$ , $N_{2} = - P (u . \nabla) u_{t}$ , $N_{3} = - P (u_{t} . \nabla) u$ , and $N_{4} = P f$ .

First, for $α = 0$ , since $div u = 0$ , we obtain after integration by parts and the Hölder inequality $\begin{array}{l} \begin{matrix} (N_{1}, u + 2 u_{t}) & = - 2 \int (u . \nabla) u u_{t} \\ ⩽ c ‖ u ‖_{L^{\infty}} ‖ \nabla u ‖_{L^{2}} ‖ u_{t} ‖_{L^{2}} ⩽ c ‖ u ‖_{L^{\infty}} Y (t), \end{matrix} \\ \begin{matrix} (N_{2}, u + 2 u_{t}) & = \int (u . \nabla) u u_{t} \\ ⩽ ‖ u ‖_{L^{\infty}} ‖ \nabla u ‖_{L^{2}} ‖ u_{t} ‖_{L^{2}} ⩽ c ‖ u ‖_{L^{\infty}} Y (t), \end{matrix} \end{array}$ and by interpolation inequalities and the Sobolev embeddings, we have $\begin{array}{rcl} (N_{3}, u + 2 u_{t}) & = & - 2 \int (u_{t} . \nabla) u u_{t} \\ ⩽ & c ‖ u_{t} ‖_{L^{4}}^{2} ‖ \nabla u ‖_{L^{2}} ⩽ c ‖ u_{t} ‖_{L^{2}}^{\frac{1}{2}} ‖ u_{t} ‖_{L^{6}}^{\frac{3}{2}} ‖ \nabla u ‖_{L^{2}} \\ ⩽ & c Y^{\frac{3}{2}} (t) . \end{array}$ Moreover, by the Young inequality, we get $\begin{array}{rcl} (N_{4}, u + 2 u_{t}) & ⩽ & ‖ f ‖_{L^{2}} (‖ u + u_{t} ‖_{L^{2}} + ‖ u_{t} ‖_{L^{2}}) \\ ⩽ & ‖ f ‖_{L^{2}} ‖ u + u_{t} ‖_{L^{2}} + c ‖ f ‖_{L^{2}}^{2} + \frac{1}{4} Y (t) . \end{array}$ We next estimate the second term of (3.2) for $| α | ⩾ 1$ .

Applying Lemma 2.1 yields $\begin{array}{rcl} (\nabla^{α} N_{1}, \nabla^{α} (u + 2 u_{t})) & ⩽ & {‖ \nabla^{α} ((u . \nabla) u) ‖}_{L^{2}} {‖ \nabla^{α} (u + 2 u_{t}) ‖}_{L^{2}} \\ ⩽ & c (‖ u ‖_{L^{\infty}} {‖ \nabla^{| α |} \nabla u ‖}_{L^{2}} + ‖ \nabla u ‖_{L^{\infty}} {‖ \nabla^{| α |} u ‖}_{L^{2}}) ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) \\ ⩽ & c (‖ u ‖_{L^{\infty}} + ‖ \nabla u ‖_{L^{\infty}}) Y (t) . \end{array}$ The Sobolev inequality implies $\begin{array}{rcl} (3.3) & ‖ \nabla u ‖_{L^{\infty}} ⩽ c ‖ \nabla u ‖_{H^{2}} ⩽ c Y^{\frac{1}{2}} (t) . \end{array}$ Hence $\begin{array}{rcl} (\nabla^{α} N_{1}, \nabla^{α} (u + 2 u_{t})) ⩽ c (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) Y (t) . \end{array}$ In order to estimate the second term $(\nabla^{α} N_{2}, \nabla^{α} (u + 2 u_{t}))$ , we write $\begin{array}{rcl} (\nabla^{α} N_{2}, \nabla^{α} (u + 2 u_{t})) & = & - \int (\nabla^{α} u . \nabla) u_{t} \nabla^{α} (u + 2 u_{t}) - \int (u . \nabla) \nabla^{α} u_{t} \nabla^{α} (u + 2 u_{t}) \\ - \sum_{| α_{1} | ⩾ 1, | α_{2} | ⩾ 1, α = α_{1} + α_{2}} \int (\nabla^{α_{1}} u . \nabla) \nabla^{α_{2}} u_{t} \nabla^{α} (u + 2 u_{t}) \\ : = & I_{1} + I_{2} + I_{3} . \end{array}$ On the one hand, by the Hölder inequality, the Sobolev injections and interpolation inequalities, we have $\begin{array}{rcl} I_{1} & ⩽ & {‖ \nabla^{α} u ‖}_{L^{6}} ‖ \nabla u_{t} ‖_{L^{3}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) \\ ⩽ & {‖ \nabla^{α} u ‖}_{H^{1}} ‖ \nabla u_{t} ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u_{t} ‖_{H^{1}}^{\frac{1}{2}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) . \end{array}$ As $| α | ⩾ 1$ , we obtain $\begin{array}{rcl} I_{1} & ⩽ & c Y^{\frac{3}{2}} (t) . \end{array}$ Performing integration by parts to the second term $I_{2}$ and using the Hölder inequality , we find $\begin{array}{rcl} I_{2} & = & \int (u . \nabla) \nabla^{α} u \nabla^{α} u_{t} ⩽ ‖ u ‖_{L^{\infty}} {‖ \nabla \nabla^{α} u ‖}_{L^{2}} {‖ \nabla^{α} u_{t} ‖}_{L^{2}} \\ ⩽ & c ‖ u ‖_{L^{\infty}} Y (t) . \end{array}$ Let us note that, $I_{3}$ exists only for $| α_{1} |, | α_{2} | \in {1, 2}$ . Then, from the Hölder inequality and inequality (3.3) $\begin{matrix} \int (\nabla^{α_{1}} u . \nabla) \nabla^{α_{2}} u_{t} \nabla^{α} (u + 2 u_{t}) ⩽ c {‖ \nabla^{α_{1}} u ‖}_{L^{\infty}} {‖ \nabla \nabla^{α_{2}} u_{t} ‖}_{L^{2}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) ⩽ c Y^{\frac{3}{2}} (t) . \end{matrix}$ Summing up, we find $\begin{matrix} I_{3} ⩽ c Y^{\frac{3}{2}} (t) . \end{matrix}$ Collecting the bounds on $I_{1}$ , $I_{2}$ and $I_{3}$ , we can write the following inequality $\begin{matrix} (\nabla^{α} N_{2}, \nabla^{α} (u + 2 u_{t})) ⩽ c (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) Y (t) . \end{matrix}$ Similarly, we decompose $N_{3}$ as follows $\begin{array}{rcl} (\nabla^{α} N_{3}, \nabla^{α} (u + 2 u_{t})) & = & - \int (\nabla^{α} u_{t} . \nabla) u \nabla^{α} (u + 2 u_{t}) - \int (u_{t} . \nabla) \nabla^{α} u \nabla^{α} (u + 2 u_{t}) \\ - \sum_{| α_{1} | ⩾ 1, | α_{2} | ⩾ 1, α = α_{1} + α_{2}} \int (\nabla^{α_{1}} u_{t} . \nabla) \nabla^{α_{2}} u \nabla^{α} (u + 2 u_{t}) \\ : = & J_{1} + J_{2} + J_{3} . \end{array}$ Using the Hölder inequality and the inequality (3.3), we obtain $\begin{array}{rcl} J_{1} & ⩽ & ‖ \nabla u ‖_{L^{\infty}} {‖ \nabla^{α} u_{t} ‖}_{L^{2}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) \\ ⩽ & c Y^{\frac{3}{2}} (t) . \end{array}$ Once again integration by parts, the Hölder inequality and the Sobolev embeddings yields $\begin{array}{rcl} J_{2} & = & - 2 \int (u_{t} . \nabla) \nabla^{α} u \nabla^{α} u_{t} ⩽ c ‖ u_{t} ‖_{L^{\infty}} {‖ \nabla \nabla^{α} u ‖}_{L^{2}} {‖ \nabla^{α} u_{t} ‖}_{L^{2}} \\ ⩽ & c ‖ u_{t} ‖_{H^{2}} {‖ \nabla \nabla^{α} u ‖}_{L^{2}} {‖ \nabla^{α} u_{t} ‖}_{L^{2}} ⩽ c Y^{\frac{3}{2}} (t) . \end{array}$ For $J_{3}$ , we get after interpolation inequalities and the Sobolev embeddings $\begin{array}{l} \int (\nabla^{α_{1}} u_{t} . \nabla) \nabla^{α_{2}} u) \nabla^{α} (u + 2 u_{t}) \\ ⩽ {‖ \nabla^{α_{1}} u_{t} ‖}_{L^{6}} {‖ \nabla \nabla^{α_{2}} u ‖}_{L^{3}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) \\ ⩽ c {‖ \nabla^{α_{1}} u_{t} ‖}_{H^{1}} {‖ \nabla \nabla^{α_{2}} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla \nabla^{α_{2}} u ‖}_{H^{1}}^{\frac{1}{2}} ({‖ \nabla^{α} u ‖}_{L^{2}} + {‖ \nabla^{α} u_{t} ‖}_{L^{2}}) . \end{array}$ Since $| α_{1} |, | α_{2} | \in {1, 2}$ , we deduce the following inequality $\begin{matrix} J_{3} ⩽ c Y^{\frac{3}{2}} (t) . \end{matrix}$ We conclude that $\begin{array}{rcl} (\nabla^{α} N_{3}, \nabla^{α} (u + 2 u_{t})) ⩽ c Y^{\frac{3}{2}} (t) . \end{array}$ Finally, applying the Hölder inequality and the Young inequality, we obtain $\begin{array}{rcl} (\nabla^{α} N_{4}, \nabla^{α} (u + 2 u_{t})) ⩽ c {‖ \nabla^{α} f ‖}_{L^{2}}^{2} + \frac{1}{4} Y (t) . \end{array}$ Collecting for $0 ⩽ | α | ⩽ 3$ all estimates, we obtain $\begin{array}{rcl} \frac{d}{d t} E (t) + (\frac{1}{2} - C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t))) Y (t) ⩽ ‖ f ‖_{L^{2}} ‖ u + u_{t} ‖_{L^{2}} + K \sum_{| α | ⩽ 3} {‖ \nabla^{α} f ‖}_{L^{2}}^{2} . \end{array}$ □

Let $T_{max}$ be the maximal existence time, we shall prove that $T_{max} = + \infty$ . For this purpose, we prove that for $0 ⩽ t ⩽ T_{max}$ $\begin{array}{rcl} (3.4) & C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) ⩽ \frac{1}{4} . \end{array}$ On the one hand, we remark that by the Sobolev injections, we have $\begin{matrix} ‖ u ‖_{L^{\infty}} ⩽ c ‖ u ‖_{H^{2}} ⩽ C (ν) E^{\frac{1}{2}} (t) . \end{matrix}$ In particular there exists a positive constant $K (ν)$ , such that $\begin{array}{rcl} (3.5) & C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) ⩽ K (ν) E^{\frac{1}{2}} (t) . \end{array}$ If we assume $K (ν) E^{\frac{1}{2}} (0) ⩽ \frac{1}{8}$ , then by continuity of the local solution $(u, u_{t})$ with respect to t, we deduce that there exists a time $T > 0$ such that for all $0 ⩽ t ⩽ T$ $\begin{matrix} K (ν) E^{\frac{1}{2}} (t) ⩽ \frac{1}{4} . \end{matrix}$ Let define $\begin{array}{rcl} (3.6) & T_{0} = sup {t, C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) ⩽ \frac{1}{4}} . \end{array}$ Assume $T_{0} < T_{max}$ , then for all $0 ⩽ t ⩽ T_{0}$ $\begin{array}{rcl} \frac{d}{d t} E (t) ⩽ ‖ f ‖_{L^{2}} ‖ u + u_{t} ‖_{L^{2}} + K \sum_{| α | ⩽ 3} {‖ \nabla^{α} f ‖}_{L^{2}}^{2} . \end{array}$ Integrating the above inequality in time between 0 and t and using the Young inequality, we obtain $\begin{array}{rcl} E (t) ⩽ E (0) + \frac{1}{2} sup_{t \in [0, T_{0}]} E (t) + C_{0} (‖ f ‖_{L^{1} ((0, + \infty), L^{2})}^{2} + ‖ f ‖_{L^{2} ((0, + \infty), H^{3})}^{2}) . \end{array}$ Therefore, $\begin{array}{rcl} (3.7) & sup_{t \in [0, T_{0}]} E (t) ⩽ 2 E (0) + 2 C_{0} (‖ f ‖_{L^{1} (L^{2})}^{2} + ‖ f ‖_{L^{2} (H^{3})}^{2}) . \end{array}$ Thanks to inequality (3.5), we find $\begin{matrix} C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) ⩽ \sqrt{2} K (ν) {(E (0) + C_{0} (‖ f ‖_{L^{1} (L^{2})}^{2} + ‖ f ‖_{L^{2} (H^{3})}^{2}))}^{\frac{1}{2}} . \end{matrix}$ If we assume that the initial data and the forcing term satisfy $\begin{matrix} (E (0) + C_{0} {(‖ f ‖_{L^{1} (L^{2})}^{2} + ‖ f ‖_{L^{2} (H^{3})}^{2})}^{\frac{1}{2}} ⩽ \frac{1}{8 K (ν)} . \end{matrix}$ Thus for all $0 ⩽ t ⩽ T_{0}$ $\begin{matrix} C (‖ u ‖_{L^{\infty}} + Y^{\frac{1}{2}} (t)) < \frac{1}{4} \end{matrix}$ which contradict (3.6). So $T_{0} ⩾ T_{max}$ and the estimate on energy $E (t)$ remains true on the whole existence interval $[0, T_{max})$ . Therefore the (HNS) equation has a global solution and the proof of Theorem 1.1 is complete.
4. Convergence

In this section, we show that the hyperbolic perturbed problem ${(HNS)}_{ε}$ is closely related to the classical Navier–Stokes problem, in the sense that the solution $u^{ε}$ of ${(HNS)}_{ε}$ converges for $ε \to 0$ to the solution of (NS).

Let v be a solution to the Helmholtz-projected Navier–Stokes equation $\begin{matrix} \{\begin{matrix} v_{t} - ν Δ v + P (v . \nabla) v = P f^{ε}, \\ div v = 0, \\ v (0, y) = v_{0} . \end{matrix} \end{matrix}$ Differentiating this equation with respect to t, we obtain $\begin{matrix} v_{t t} - ν Δ v_{t} + P (v . \nabla) v_{t} + P (v_{t} . \nabla) v = P f_{t}^{ε} . \end{matrix}$ Multiplying the above equation with ε and adding the original equation yields $\begin{matrix} \{\begin{matrix} ε v_{t t} - ν Δ v + v_{t} - ε ν Δ v_{t} + P (v . \nabla) v + ε P (v . \nabla) v_{t} + ε P (v_{t} . \nabla) v = P f^{ε} + ε P f_{t}^{ε}, \\ div v = 0, \\ v (0, y) = v_{0}, v_{t} (0, y) = ν Δ v_{0} - P (v_{0} . \nabla) v_{0} + P f^{ε} (0, y) : = v_{1} . \end{matrix} \end{matrix}$ Let $w = u^{ε} - v$ . Therefore, the equation satisfied by w can be written as $\begin{matrix} (4.1) & \{\begin{matrix} ε w_{t t} - ν Δ w + w_{t} + ε ν Δ v_{t} \\ = - P ((w - ε w_{t}) . \nabla) v - P ((u^{ε} - ε u_{t}^{ε}) . \nabla) w - ε P (u^{ε} . \nabla) w_{t} - ε P (w . \nabla) v_{t} - ε P f_{t}^{ε}, \\ div w = 0, \\ w (0, y) = u_{0}^{ε} - v_{0}, w_{t} (0, y) = u_{1}^{ε} - v_{1} . \end{matrix} \end{matrix}$ Let us define the energy $\begin{matrix} E_{w}^{ε} (t) = ‖ w + ε w_{t} ‖_{L^{2}}^{2} + ε^{2} ‖ w_{t} ‖_{L^{2}}^{2} + ν ε ‖ \nabla w ‖_{L^{2}}^{2} . \end{matrix}$ In order to prove the convergence of w, we state the following lemma.

Lemma 4.1.

Let $T > 0$ . There exists a positive constant C such that for all $0 ⩽ t ⩽ T$ $\begin{array}{l} E_{w}^{ε} (t) + \int_{0}^{t} (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}) d s \\ ⩽ E_{w}^{ε} (0) + C \int_{0}^{t} (‖ \nabla v ‖_{L^{\infty}} + ε^{\frac{1}{2}} ‖ \nabla v_{t} ‖_{L^{3}}) E_{w}^{ε} (s) d s \\ (4.2) & + C ε^{2} \int_{0}^{t} (‖ \nabla v_{t} ‖_{L^{2}}^{2} + ε ‖ Δ v_{t} ‖_{L^{2}}^{2} + {‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2}}^{2}) d s . \end{array}$

Proof.

Taking the $L^{2}$ inner product of the equation (4.1) with $w + 2 ε w_{t}$ , we obtain $\begin{matrix} \frac{1}{2} \frac{d}{d t} E_{w}^{ε} (t) + ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2} = \sum_{i = 1}^{6} (B_{i}, w + ε w_{t}), \end{matrix}$ where $B_{1} = - ε ν Δ v_{t}$ , $B_{2} = - P ((w - ε w_{t}) . \nabla) v$ , $B_{3} = - P ((u^{ε} - ε u_{t}^{ε}) . \nabla) w$ , $B_{4} = - ε P (u^{ε} . \nabla) w_{t}$ , $B_{5} = - ε P (w . \nabla) v_{t}$ and $B_{6} = - ε P f_{t}^{ε}$ .

First, using integration by parts and the Hölder inequality followed by the Young inequality, we get $\begin{array}{rcl} (B_{1}, w + 2 ε w_{t}) & = & ε ν \int \nabla v_{t} \nabla w - 2 ε^{2} ν \int Δ v_{t} w_{t} \\ ⩽ & c (ε^{2} ‖ \nabla v_{t} ‖_{L^{2}}^{2} + ε^{3} ‖ Δ v_{t} ‖_{L^{2}}^{2}) + \frac{1}{8} (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}), \end{array}$ and $\begin{array}{rcl} (B_{2}, w + 2 ε w_{t}) & ⩽ & ‖ \nabla v ‖_{L^{\infty}} (‖ w ‖_{L^{2}} + ε ‖ w_{t} ‖_{L^{2}}) ‖ w + ε w_{t} ‖_{L^{2}} \\ ⩽ & c ‖ \nabla v ‖_{L^{\infty}} E_{w}^{ε} (t) . \end{array}$ Analogously, since $div w = 0$ , we obtain $\begin{array}{rcl} (B_{3}, w + 2 ε w_{t}) & = & - 2 ε \int ((u^{ε} - ε u_{t}^{ε}) . \nabla) w w_{t} \\ ⩽ & c \sqrt{ε} ({‖ u^{ε} ‖}_{L^{\infty}} + ε {‖ u_{t}^{ε} ‖}_{L^{\infty}}) (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}) . \end{array}$ Similarly, using the fact that $div u_{ε} = 0$ , we get $\begin{array}{rcl} (B_{4}, w + 2 ε w_{t}) & = & ε \int (u^{ε} . \nabla) w w_{t} \\ ⩽ & c \sqrt{ε} {‖ u^{ε} ‖}_{L^{\infty}} (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}) . \end{array}$ Applying the Hölder inequality and the Sobolev embeddings, we find $\begin{array}{rcl} (B_{5}, w + 2 ε w_{t}) & ⩽ & ε ‖ \nabla v_{t} ‖_{L^{3}} ‖ w ‖_{L^{6}} ‖ w + ε w_{t} ‖_{L^{2}} \\ ⩽ & c \sqrt{ε} ‖ \nabla v_{t} ‖_{L^{3}} E_{w}^{ε} (t) . \end{array}$ Once again, we apply the Hölder inequality, the Sobolev embeddings and the Young inequality to the last term yields $\begin{array}{rcl} (B_{6}, w + 2 ε w_{t}) & ⩽ & ε {‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}} ‖ w ‖_{{\dot{H}}^{1}} + 2 ε^{2} {‖ f_{t}^{ε} ‖}_{L^{2}} ‖ w_{t} ‖_{L^{2}} \\ ⩽ & c ε^{2} ({‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2}}^{2}) + \frac{1}{8} (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}) . \end{array}$ Collecting the above estimates, we deduce that there exist positive constants K and C such that $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} E_{w}^{ε} (t) + ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2} & ⩽ & (\frac{1}{4} + K \sqrt{ε} ({‖ u^{ε} ‖}_{L^{\infty}} + ε {‖ u_{t}^{ε} ‖}_{L^{\infty}})) (ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2}) \\ + C (‖ \nabla v_{t} ‖_{L^{\infty}} + \sqrt{ε} ‖ \nabla v_{t} ‖_{L^{3}}) E_{w}^{ε} (t) \\ + C ε^{2} (‖ \nabla v_{t} ‖_{L^{2}}^{2} + ε ‖ Δ v_{t} ‖_{L^{2}}^{2} + {‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2}}^{2}) . \end{array}$ Its remains to show that $K \sqrt{ε} (‖ u^{ε} ‖_{L^{\infty}} + ε ‖ u_{t}^{ε} ‖_{L^{\infty}}) ⩽ \frac{1}{4}$ . For this purpose, we remark that by scaling $\begin{matrix} \sqrt{ε} ({‖ u^{ε} ‖}_{L^{\infty}} + ε {‖ u_{t}^{ε} ‖}_{L^{\infty}}) = ‖ u ‖_{L^{\infty}} + ‖ u_{t} ‖_{L^{\infty}} . \end{matrix}$ Thanks to the Sobolev embeddings and the energy estimate (3.7), we get $\begin{matrix} K \sqrt{ε} ({‖ u^{ε} ‖}_{L^{\infty}} + ε {‖ u_{t}^{ε} ‖}_{L^{\infty}}) ⩽ C {(E (0) + (‖ f ‖_{L^{1} (L^{2})}^{2} + ‖ f ‖_{L^{2} (H^{3})}^{2}))}^{\frac{1}{2}}, \end{matrix}$ we claim the desired inequality since the initial data and the forcing term satisfy $C {(E (0) + (‖ f ‖_{L^{1} (L^{2})}^{2} + ‖ f ‖_{L^{2} (H^{3})}^{2}))}^{\frac{1}{2}} ⩽ \frac{1}{4}$ . Hence, $\begin{array}{rcl} \frac{d}{d t} E_{w}^{ε} (t) + ν ‖ \nabla w ‖_{L^{2}}^{2} + ε ‖ w_{t} ‖_{L^{2}}^{2} & ⩽ & C (‖ \nabla v_{t} ‖_{L^{\infty}} + \sqrt{ε} ‖ \nabla v_{t} ‖_{L^{3}})) E_{w}^{ε} (t) \\ + C ε^{2} (‖ \nabla v_{t} ‖_{L^{2}}^{2} + ε ‖ Δ v_{t} ‖_{L^{2}}^{2} + {‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2}}^{2}) . \end{array}$ Integrating in time between 0 and t the energy estimate (4.2) is proved. □

Next, we want to prove the convergence of w. If we assume that there exists M such that $\begin{matrix} ε^{- 1} {‖ u_{0}^{ε} - v_{0} ‖}_{L^{2}} + ε^{- \frac{1}{2}} {‖ \nabla (u_{0}^{ε} - v_{0}) ‖}_{L^{2}} + {‖ u_{1}^{ε} - v_{1} ‖}_{L^{2}} ⩽ M, \end{matrix}$ then there exists a positive constant C such that $E_{w}^{ε} (0) ⩽ C ε^{2}$ .

Applying the Gronwall Lemma to (4.2), we find $\begin{array}{rcl} E_{w}^{ε} (t) & ⩽ & C ε^{2} (1 + \int_{0}^{t} (‖ \nabla v_{t} ‖_{L^{2}}^{2} + ε ‖ Δ v_{t} ‖_{L^{2}}^{2} + {‖ f_{t}^{ε} ‖}_{{\dot{H}}^{- 1}}^{2} + ε {‖ f_{t}^{ε} ‖}_{L^{2}}^{2}) d s) \\ \times exp (C \int_{0}^{t} ‖ \nabla v ‖_{L^{\infty}} + ε^{\frac{1}{2}} ‖ \nabla v_{t} ‖_{L^{3}} d s) . \end{array}$ Thus Theorem 1.2 is proved.

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