On the Jordan-Moore-Gibson-Thompson wave equation in hereditary fluids with quadratic gradient nonlinearity
aa r X i v : . [ m a t h . A P ] M a y ON THE JORDAN–MOORE–GIBSON–THOMPSON WAVEEQUATION IN HEREDITARY FLUIDS WITH QUADRATICGRADIENT NONLINEARITY
VANJA NIKOLI ´C ∗ AND BELKACEM SAID-HOUARI
Abstract.
We prove global solvability of the third-order in time Jordan–More–Gibson–Thompson acoustic wave equation with memory in R n , where n ≥ Introduction
The present paper focuses on the analysis of a third-order in time integro-differential equation arising in nonlinear acoustic wave propagation through viscousfluids with memory. We are motivated in this study by an increasing number of ap-plications of high-frequency sound waves in medicine and industry [11, 12, 29, 30].The propagation of ultrasound waves is naturally nonlinear and, therefore, the the-ory of nonlinear partial differential equations can offer a valuable insight into theultrasonic wave behavior.Of particular relevance here are the ultrasonic waves in relaxing hereditary me-dia. The relaxation mechanisms can occur, for example, if there is an impurity inthe fluid and they are known to introduce memory effects into propagation. Theacoustic pressure then depends on the medium density at all previous times, result-ing in a nonlocal in time wave equation; see, for example, the recent book [13]. Inparticular, we investigate the following nonlinear acoustic wave equation: τ ψ ttt + αψ tt − c ∆ ψ − b ∆ ψ t + Z t g ( s )∆ ψ ( t − s ) d s = (cid:0) kψ t + |∇ ψ | (cid:1) t . We refer to Section 2 below for its physical background. Our work here extends theanalysis in [34] by taking into account local effects in nonlinear sound propagation,which leads to a quadratic gradient nonlinearity in the model above. The energyarguments of [34] are then not adequate to capture such nonlinear effects. Instead,we have to involve higher-order energies of our system and devise new estimates.In addition, we extend the analysis in R n to hold for all n ≥ Mathematics Subject Classification.
Key words and phrases. nonlinear acoustics, nonlocal wave equation, relaxing media, memorykernel, gradient nonlinearity. ∗ Corresponding author: Vanja Nikoli´c, [email protected]. acoustic propagation in relaxing hereditary media and the relevant related work.Section 3 then collects theoretical results that are helpful in later proofs. In Sec-tion 4, we rewrite the problem as a first-order evolution equation and discuss thesemigroup solution of the linearization. This serves as a basis in Section 5 to provethe local solvability of the nonlinear problem. Section 6 is devoted to derivingenergy estimates that are uniform in time and thus crucial for proving global well-posedness. Finally, in Section 7, we extend the existence result to T = ∞ .2. Modeling and previous work
Nonlinear acoustics studies sound waves of sufficiently large amplitudes, whichmakes using the full Navier–Stokes system of governing equations in fluid dynamicsnecessary. The equations connect the following quantities: • the pressure u , split into its mean and alternating part u = u + u ′ with ∇ u = 0; • the velocity v = v + v ′ , which is assumed to be irrotational; • the mass density ̺ , where ̺ = ̺ + ̺ ′ with ̺ ,t = 0; • the specific entropy η , where η = η + η ′ ; • the temperature θ , where θ = θ + θ ′ ; • the heat flux q .The scalar field u ′ is called the acoustic pressure and vector field v ′ the acousticparticle velocity. The governing equations include the conservation of momentum(the Navier–Stokes equation), mass, and energy: ̺ ∂ v ′ ∂t + ̺ ( v · ∇ ) v + ∇ u ′ = (cid:18) µ v η v (cid:19) ∆ v ′ ,∂̺∂t + ∇ · ( ̺ v ) = 0 ,̺ (cid:18) ∂E∂t + ( ∇ · v ) E (cid:19) + u ∇ · v = K ∆ θ ′ + η ( ∇ · v ) + 12 ( ∂ i v j + ∂ j v i − ∇ · v δ ij ) , where K is the heat conductivity, µ v is the shear viscosity, η v denotes the bulkviscosity, and δ ij is the Kroneker delta. The state equation relates the pressure anddensity within a fluid: u = u ( ̺, η ) . By expanding it in a Taylor series around the equilibrium state ( ̺ , η ), one arrivesat u = u + ̺ (cid:16) ∂u∂̺ (cid:17) ̺ = ̺ ,η ̺ − ̺ ̺ + 12 ̺ (cid:16) ∂ u∂̺ (cid:17) ̺ = ̺ ,η (cid:18) ̺ − ̺ ̺ (cid:19) + (cid:16) ∂u∂η (cid:17) ̺,η = η ( η − η ) + . . . This equation can be rewritten as u − u = A ̺ − ̺ ̺ + B (cid:16) ̺ − ̺ ̺ (cid:17) + (cid:16) ∂u∂η (cid:17) ̺,η = η ( η − η ) + . . . , with the coefficients A = ̺ (cid:16) ∂u∂̺ (cid:17) ̺ = ̺ ,η ≡ ̺ c , B = ̺ (cid:16) ∂ u∂̺ (cid:17) ̺ = ̺ ,η , HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 3 where c > u − u = c ( ̺ − ̺ ) + c ̺ B A ( ̺ − ̺ ) + (cid:16) ∂u∂η (cid:17) ̺,η = η ( η − η ) + . . . The ratio
B/A indicates the nonlinearity of the equation of state for a given fluid.Lastly, to close the system, the classical Fourier law of heat conduction is employedin the equation for the conservation of energy: q = − K ∇ θ, The so-called weakly nonlinear acoustic modeling introduces and studies approx-imations of these governing equations. We refer to [16] for a survey on such models.Since the fluid is irrotational, a scalar acoustic velocity potential can be introducedas v = −∇ ψ,u − u = − ̺ψ t , and these weakly nonlinear acoustic equations are typically expressed in terms of ψ . The deviations of ̺ , u , η , and θ from their equilibrium values are assumed to besmall. By neglecting all third and higher order terms in the deviations when com-bining the governing equations, one obtains the classical Kuznetsov wave equation: ψ tt − c ∆ ψ − δ ∆ ψ t = (cid:18) c B A ( ψ t ) + |∇ ψ | (cid:19) t . For a detailed derivation, we refer the reader to [6, 7, 21–23]. The coefficient δ > |∇ ψ | ≈ c ψ t , and one obtains the simpler Westervelt model [40].It is known, however, that the Fourier law used in the derivation of the Kuznetsovequation predicts an infinite speed of heat propagation [27]. Indeed, the strong δ damping in the model is proven to lead to a parabolic-like behavior with anexpo-nential decay of energy [17, 31]. The Maxwell–Cattaneo temperature law can beused instead to avoid this paradox for acoustic waves: τ q t + q = − K ∇ θ, where τ > τ ψ ttt + ψ tt − c ∆ ψ − b ∆ ψ t = (cid:18) c B A ( ψ t ) + |∇ ψ | (cid:19) t , often called the Jordan–Moore–Gibson–Thompson (JMGT) equation (of Kuznetsov-type); see [15]. The coefficient b > b = δ + τ c . This model is mathematically well-studied in terms of well-posedness and regularityof solutions both in bounded domains [20] and in R [38]. Furthermore, in [20], it V. NIKOLI´C & B. SAID-HOUARI is rigorously justified that the limit of (2.2) as τ → τ ψ ttt + αψ tt − c ∆ ψ − b ∆ ψ t = 0 , where α > b >
0, the results of [18] show that thelinear dynamics of this model is described by a strongly continuous semigroup. Thesemigroup is exponentially stable provided that the so-called subcritical conditionholds: αb − τ c > . (2.3)Interestingly, the semigroup is conservative in the critical case αb = τ c – the energyis conserved despite the presence of sound diffusivity. The linear MGT equationhas been extensively studied lately; see, for example, [3–5, 28, 36, 37].When relaxation processes occur in high-frequency waves, the acoustic pressurecan depend on the medium density at all prior times. These processes happen, forexample, when there is an impurity in the fluid. The pressure-density relation theninvolves a memory term: u − u = c ( ̺ − ̺ ) + c ̺ B A ( ̺ − ̺ ) + (cid:16) ∂u∂η (cid:17) ̺,η = η ( η − η ) − Z t g ( t − s ) ̺ ′ ( s ) d s ;cf. [13]. The function g is the relaxation memory kernel related to the occurringrelaxation mechanism.Such memory effects in nonlinear wave propagation motivate our work in thepresent paper. Our object of study is the non-local JMGT equation with quadraticgradient nonlinearity given by(2.4) τ ψ ttt + αψ tt − c ∆ ψ − b ∆ ψ t + Z t g ( s )∆ ψ ( t − s ) d s = (cid:0) kψ t + |∇ ψ | (cid:1) t . The constant k ∈ R indicates the nonlinearity of the equation. In the present work,we consider memory that involves only the acoustic velocity potential and not itstime derivatives, which is sometimes regarded as memory of type I.The linear model associated with the JMGT equation with memory in the pressueform τ u ′ ttt + αu ′ tt − c ∆ u ′ − b ∆ u ′ t + Z t g ( s )∆ z ( t − s ) d s = 0 . has also been subject to extensive study recently. In [26], the effects of differentmemories on the stability of the equation are investigated. In particular, the authorsanalyzed the equation in the pressure form with z = u ′ (memory of type I), z = u ′ t (type II), or z = u ′ + u ′ t (type III). If the memory kernel g decays exponentially,the same holds for the solution for all three types of memory, provided that thenon-critical condition (2.3) holds. This result is extended in [25] to include not onlyan exponential decay rate of the memory kernel.The critical case αb = τ c , where the presence of memory is essential, is analyzedin [10] with memory of type I (depending only on the pressure). With a strictlypositive self-adjoint linear operator A in place of − ∆, the problem is exponentiallystable if and only if A is a bounded operator. In the case of an unbounded operator HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 5 A , the corresponding energy decays polynomially with the rate 1 /t for regular initialdata.Neglecting local nonlinear effects in (2.4) via relation (2.1) leads to a Westervelt-type JMGT equation with memory. In [24], this third-order nonlinear equation isstudied in the pressure form in the critical case αb = τ c in the presence of memorytype III (depending on u ′ and u ′ t ). It is shown that an appropriate adjustment ofthe memory kernel allows having solutions globally in time for sufficiently small andregular initial data. Global solvability for the equation in potential form in R forsmall and regular data is proven in [34] in the non-critical case and with memorydepending only on ψ .In relaxing media, the memory kernel is often given by the exponential function g ( s ) = mc exp ( − s/τ ) , where m is the relaxation parameter; see [33, Chapter 1] and [24, Section 1]. Moti-vated by this, we make the following assumptions on the relaxation kernel through-out the paper; see also [10, 34]. Assumptions on the memory kernel.
The memory kernel is assumed to satisfythe following conditions: (G1) g ∈ W , ( R + ) and g ′ is almost continuous on R + = (0 , + ∞ ) . (G2) g ( s ) ≥ for all s > and < Z ∞ g ( s ) d s < c . (G3) There exists ζ > , such that the function g satisfies the differential inequal-ity given by g ′ ( s ) ≤ − ζg ( s ) for every s ∈ (0 , ∞ ) . (G4) It holds that g ′′ ≥ almost everywhere. Auxiliary theoretical results
For future use in the analysis of equation (2.4), we recall here several helpfultheoretical results and set the notation. We choose to work in the history frameworkof Dafermos [8], following previous work on acoustic equations with memory; see,for example, [10]. This is achieved by introducing the auxiliary past-history variable η = η ( t, s ) for t ≥
0, defined as η ( s ) = ( ψ ( t ) − ψ ( t − s ) , < s ≤ t,ψ ( t ) , s > t. If we choose η | t =0 = ψ ( x ), we can rewrite our problem as τ ψ ttt + αψ tt − b ∆ ψ t − c g ∆ ψ − Z ∞ g ( s )∆ η ( s ) d s = 2 kψ t ψ tt + 2 ∇ ψ · ∇ ψ t ,η t ( x, s ) + η s ( x, s ) = ψ t ( x, t ) , where we have introduced the modified speed c g = c − Z ∞ g ( s ) d s. V. NIKOLI´C & B. SAID-HOUARI
Thanks to our assumptions on the memory kernel, we have c g >
0. The equationsare supplemented with the initial data ψ ( x,
0) = ψ ( x ) , ψ t ( x,
0) = ψ ( x ) , ψ tt ( x,
0) = ψ ( x ) . Setting α = 1. We can take α = 1 without the loss of generality because we canalways re-scale other coefficients in the equation. The subcritical condition thenreads as b > τ c , which is equivalent to requiring the sound diffusivity δ to be positive since b = δ + τ c . The critical case corresponds to b = τ c . We always require the presenceof memory via the assumption τ c > τ c g ; that is, R ∞ g ( s ) d s > Notation.
In the present work, the constant C always stands for a genericpositive constant that does not depend on time, and it may have different value ondifferent occasions. We write x . y instead of x ≤ Cy .3.2. Helpful inequalities.
We collect here several inequalities that are used through-out the proofs. We often rely on the following endpoint Sobolev embedding:(3.1) k ψ k L nn − . k∇ ψ k L , which holds provided that n ≥
3; cf. [1]. Going forward, we therefore assume n ≥ k f gh k L . k f k L k g k L n k h k L nn − together with the embedding H n − ( R n ) ֒ → L n ( R n ) . (3.3)Furthermore, when estimating the nonlinear terms in the equation, we will rely onthe embedding H r ( R n ) ֒ → L ∞ ( R n ) , r > n/ . We will work with the space-differentiated equation as well, and so commutatorestimates are particularly helpful. We introduce the commutator notation by(3.4) [
A, B ] = AB − BA.
It helps to note that ∂ κ ( AB ) = [ ∂ κ , A ] B + A∂ κ B, κ ≥ . We have the following estimates.
Lemma 3.1 (see Lemma 4.1 in [14].) . Let ≤ p, q, r ≤ ∞ and /p = 1 /q + 1 /r .Then we have (3.5) k∇ κ ( f g ) k L p . k f k L q k∇ κ g k L r + k g k L q k∇ κ f k L r , κ ≥ , and the commutator estimate k [ ∇ κ , f ] g k L p = k∇ κ ( f g ) − f ∇ κ g k L p . k∇ f k L q k∇ κ − g k L r + k g k L q k∇ κ f k L r , κ ≥ . Finally, the following lemma will be useful in our energy arguments when provingglobal well-posedness.
HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 7
Lemma 3.2 (see Lemma 3.7 in [39]) . Let M = M ( t ) be a non-negative continuousfunction such that M ( t ) ≤ C + C M ( t ) κ holds in some interval containing , where C and C are positive constants and κ > . If M (0) ≤ C and C C / ( κ − < (1 − /κ ) κ − / ( κ − , then in the same interval it holds M ( t ) < C − /κ . The first-order system and the linearization
We can see our equation as a first-order in time system by introducing v = ψ t , w = ψ tt . The JMGT equation can then be restated as(4.1) ψ t = v,v t = w,τ w t = − w + c g ∆ ψ + b ∆ v + Z ∞ g ( s )∆ η ( s ) d s + 2( kvw + ∇ ψ · ∇ v ) ,η t = v − η s , with the initial data given by(4.2) ( ψ, v, w, η ) | t =0 = ( ψ , ψ , ψ , ψ ) . We then adapt the functional framework of [10] to our setting. To work in thepast-history framework of Dafermos, we introduce the weighted L -spaces, L g = L g ( R + , L ( R n ))with three types of weights: ˜ g ∈ { g, − g ′ , g ′′ } . The space is endowed with the innerproduct ( η, ˜ η ) L , ˜ g = Z ∞ ˜ g ( s ) ( η ( s ) , ˜ η ( s )) L ( R n ) d s for η, ˜ η ∈ L g . The corresponding norm is k η k L , ˜ g = Z ∞ ˜ g ( s ) k η ( s ) k L d s, Let n ≥
3. In order to formulate our results, for an integer m ≥
1, we introducethe Hilbert spaces H m − = { ψ : ∇ ψ, . . . , ∇ ( m ) ψ ∈ L ( R n ) } × H m ( R n ) × H m − ( R n ) × M m , where M m = { η : ∇ η, . . . , ∇ ( m ) η ∈ L − g ′ } . The corresponding norm is given by(4.3) k Ψ k H m − = k∇ ψ k H m − + k v k H m + k w k H m − + k∇ η k H m − , − g ′ . We recall that for n ≥
3, the endpoint Sobolev embedding (3.1) holds in R n , so k · k H m − is indeed a norm. V. NIKOLI´C & B. SAID-HOUARI
To further arrive at an initial-value problem for a first-order evolution equation,we set Ψ = ( ψ, v, w, η ) T with Ψ = Ψ(0). Moreover, we define the operator A as A ψvwη = vw − τ w + c g τ ∆ ψ + bτ ∆ v + τ Z ∞ g ( s )∆ η ( s ) d sv + T η with the domain D ( A ) = ( ψ, v, w, η ) T ∈ H m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ∈ H m ( R n ) ,c g τ ∆ ψ + bτ ∆ v + 1 τ Z ∞ g ( s )∆ η ( s ) ∈ H m − ( R n ) ,η ∈ D ( T ) . The linear operator T above is given by T η = − η s , and has the domain D ( T ) = { η ∈ M m (cid:12)(cid:12) η s ∈ M m , η ( s = 0) = 0 } . We can then formally see Ψ as the solution to(4.4) dd t Ψ( t ) = A Ψ( t ) + F (Ψ , ∇ Ψ) , t > , Ψ(0) = Ψ , with the nonlinear term given by(4.5) F (Ψ , ∇ Ψ) = 1 τ [0 , , kvw + 2 ∇ ψ · ∇ v, T . In the spirit of [38], our plan is to prove local solvability of (4.4) by introducing themapping T (Φ) = e t A Ψ + Z t e ( t − r ) A F (Φ , ∇ Φ)( r ) d r on a suitable ball in a Banach space. If we can employ the Banach fixed-point on T , the unique fixed-point of this mapping is the mild solution of (4.4).To begin with, we prove that A generates a linear C -semigroup. The prooffollows by adapting the arguments from [2] based on the Lumer-Phillips theorem.To this end we first introduce an equivalent scalar product and norm on the space H m − adapted to fit our particular problem.For any vectors Ψ = ( ψ, v, w, η ) and ¯Ψ = ( ¯ ψ, ¯ v, ¯ w, ¯ η ) in H m − , we define the HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 9 scalar product(Ψ , ¯Ψ) H m − = m − X κ =0 (cid:26) c g ( ∇ κ +1 ( ψ + τ v ) , ∇ κ +1 ( ¯ ψ + τ ¯ v )) L + τ ( b − τ c g )( ∇ κ +1 v, ∇ κ +1 ¯ v ) L + τ ( b − τ c g )( ∇ κ v, ∇ κ ¯ v ) L + ( ∇ κ ( v + τ w ) , ∇ κ (¯ v + τ ¯ w )) L + τ ( ∇ κ +1 η, ∇ κ +1 ¯ η ) L , − g ′ + ( ∇ κ +1 η, ∇ κ +1 ¯ η ) L ,g + τ Z R n (cid:18) ( ∇ κ +1 η, ∇ κ +1 ¯ v ) L ,g + ( ∇ κ +1 ¯ η, ∇ κ +1 v ) L ,g (cid:19) d x (cid:27) . The corresponding norm is given by ||| Ψ ||| H m − = m − X κ =0 (cid:26) c g k∇ κ +1 ( ψ + τ v ) k L + τ ( b − τ c g ) k∇ κ +1 v k L + τ ( b − τ c g ) k∇ κ v k L + k∇ κ ( v + τ w ) k L + τ k∇ κ +1 η k L , − g ′ + k∇ κ +1 η k L ,g + 2 τ Z R n ( ∇ κ +1 η, ∇ κ +1 v ) L ,g d x (cid:27) . We now prove that this problem-specific norm is equivalent to the standard norm(4.3) on H m − . Proposition 4.1.
Let b ≥ τ c > τ c g and m ≥ . Assume that n ≥ . There existpositive constants C and C such that (4.6) C k Ψ k H m − ≤ ||| Ψ ||| H m − ≤ C k Ψ k H m − for all Ψ ∈ H m − .Proof. The proof follows analogously to the proof of [10, Lemma 3.1], so we onlyprove the more involved left-hand side inequality here.For each κ = 0 , , . . . , m −
1, we have by Young’s inequality (cid:12)(cid:12)(cid:12)(cid:12) τ Z R n Z ∞ g ( s ) ∇ κ +1 η ( s ) · ∇ κ +1 v d s d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( c − c g ) ε + 1 k∇ κ +1 v k L + ( ε + 1) Z ∞ g ( s ) k∇ κ +1 η ( s ) k L d s, for all ε >
0. We recall assumption (G3) on the relaxation kernel g to derive2 τ Z R n Z ∞ g ( s ) ∇ κ +1 η ( s ) · ∇ κ +1 v d s d x ≥ − τ ( c − c g ) ε + 1 k∇ κ +1 v k L − k∇ κ +1 η k L ,g − εζ Z ∞ ( − g ′ ( s )) k∇ κ +1 η ( s ) k L d s. We know that c > c g and we can take ε < τ ζ to obtain(4.7) c g k∇ κ +1 ( ψ + τ v ) k L + τ ( b − τ c g ) k∇ κ +1 v k L + τ ( b − τ c g ) k∇ κ v k L + k∇ κ ( v + τ w ) k L + τ k∇ κ +1 η k L , − g ′ + k∇ κ +1 η k L ,g + 2 τ Z R n ( ∇ κ +1 η, ∇ κ +1 v ) L ,g d x ≥ c g k∇ κ +1 ( ψ + τ v ) k L + (cid:2) τ ( b − τ c g ) − τ ( c − c g ) / ( ε + 1) (cid:3) k∇ κ +1 v k L + k∇ κ ( v + τ w ) k L + ( τ − ε/ζ ) k∇ κ +1 η k L , − g ′ . Note that the condition τ ( b − τ c g ) − τ ( c − c g ) ε + 1 > ετ ( b − τ c g ) + τ ( b − τ c ) >
0, which holds true under the assumptionsof the theorem. It remains to observe that(4.8) c g k∇ κ +1 ( ψ + τ v ) k L + τ ( b − τ c g )( k∇ κ +1 v k L + k∇ κ v k L )+ k∇ κ ( v + τ w ) k L ≥ ˜ C ( k∇ κ +1 ψ k L + k∇ κ +1 v k L + k∇ κ v k L + k∇ κ w k L ) , for some positive constant ˜ C that depends on τ , c g , and b . The left-hand sideinequality in (4.6) then follows by summing (4.7) over κ = 0 , , . . . , m − (cid:3) Theorem 4.1.
Let b ≥ τ c > τ c g and m ≥ . Assume that n ≥ . Then the linearoperator A is the infinitesimal generator of a linear C -semigroup S ( t ) = e t A : H m − → H m − . Proof.
The proof is based on the Lumer-Phillips theorem applied to a boundedperturbation of the operator A ; see [35, Chapter 1.4]. To this end, we introducethe operator B by B ψvwη = − τ ( b − τ c g ) v and set A B = A + B ; see [2, Theorem 2.4] for a similar approach. Since B is abounded linear operator on H m − , if A B generates a C semigroup on H m − , thenso does A = A B − B ; cf. [35, Theorem 1.1].We first wish to prove that A B is dissipative:( A B Ψ , Ψ) H m − ≤ ∈ D ( A ) . HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 11
For a given Ψ ∈ D ( A ), it is straightforward to check that the following identityholds:( A B Ψ , Ψ) H m − = m − X κ =0 (cid:18) − ( b − τ c ) k∇ κ +1 v k L − τ ( b − τ c g ) k∇ κ v k L + Z ∞ ( g − τ g ′ ) ∇ κ +1 η ( s ) · ∇ κ +1 T η ( s ) d s + Z ∞ g ( s ) ∇ κ +1 T η ( s ) · ∇ κ +1 v d s − Z ∞ g ′ ( s ) ∇ κ +1 η ( s ) · ∇ κ +1 v d s (cid:19) . We can simplify this expression by noting that Z ∞ g ( s ) ∇ κ +1 T η ( s ) · ∇ κ +1 v d s = Z ∞ g ′ ( s ) ∇ κ +1 η ( s ) · ∇ κ +1 v d s. Therefore, we have( A B Ψ , Ψ) H m − = m − X κ =0 (cid:18) − ( b − τ c ) k∇ κ +1 v k L − ( b − τ c g ) k∇ κ v k L + Z ∞ ( g ′ − τ g ′′ ) k∇ κ +1 η ( s ) k L d s (cid:19) ≤ , where the last inequality holds thanks to the assumptions on the memory kernel.To be able to employ the Lumer-Phillips theorem, it remains to prove thatRan(I − A B ) = H m − . In other words, for a given F = ( f, g, h, p ) ∈ H m − , we have to prove existence of aunique solution Ψ = ( ψ, v, w, η ) ∈ D ( A ) to the equationΨ − A B Ψ = F. We can write this equation component-wise as the following system:(4.9) ψ = v + f,v = w + g,w = − τ w + τ c g ∆ ψ + τ b ∆ v + τ Z ∞ g ( s )∆ η ( s ) d s − τ ( b − τ c g ) v + h,η = v − η s + p. Solving the last equation and using η ( s = 0) = 0 yields η ( s ) = (1 − e − s ) v + Z s e − ( s − y ) p ( y ) d y. Combining the equations in the system results in an elliptic problem for v :(4.10) − ν ∆ v + σv = q, where the coefficients and the source term are given by ν = b + c g + Z ∞ g ( s )(1 − e − s ) d s > ,σ = 1 + τ + ( b − τ c g ) ,q = c g ∆ f + (1 + τ ) g + τ h + Z ∞ g ( s ) Z s e − ( s − y ) ∆ p ( y ) d y d s. To prove that the elliptic equation (4.10) admits a unique solution v ∈ H m ( R n ), wenote that k q k H m − . k ∆ f k H m − + k g k H m − + k h k H m − + Z ∞ g ( s ) Z s e − ( s − y ) k ∆ p ( y ) k H m − d y d s. We can further estimate the last term on the right as follows:(4.11) Z ∞ g ( s ) Z s e − ( s − y ) k ∆ p ( y ) k H m − d y d s ≤ q c − c g Z ∞ g ( s ) (cid:18)Z s e − ( s − y ) k ∆ p ( y ) k H m − d y (cid:19) d s ! / ≤ s c − c g ζ (cid:18)Z ∞ − g ′ ( s ) k ∆ p ( y ) k H m − d s (cid:19) / . The claim then follows by the Lax-Milgram theorem. From (4.9), we further have ψ = v + f ∈ { ψ : ∇ ψ, . . . , ∇ m ψ ∈ L ( R n ) } and w = v − g ∈ H m ( R n ). On accountof equation (4.10), it follows that k∇ η k H m − , − g ′ . k∇ v k H m − , − g ′ + k∇ p k H m − , − g . k∇ v k H m − + k∇ p k H m − , − g . Therefore also T η = η − v − p ∈ M m . Moreover, from (4.9), we have η (0) = 0.Finally, 1 τ c g ∆ ψ + 1 τ b ∆ v + 1 τ Z ∞ g ( s )∆ η ( s ) d s = w + 1 τ w + 1 τ ( b − τ c g ) v − h ∈ H m − ( R n ) . This completes the proof. (cid:3) Short-time existence for the JMGT equation
We claim that a unique solution to our nonlinear problem exists for sufficientlyshort final time. Since there is a quadratic gradient nonlinearity in the JMGTequation, we will need a bound on k∇ ψ k L ∞ and k∇ v k L ∞ in the upcoming estimates.We intend to employ the Sobolev embedding ∇ ψ ( t ) , ∇ v ( t ) ∈ H r ( R n ) ֒ → L ∞ ( R n ) , r > n/ . This leads to the assumption m > n/ H m − . Theorem 5.1.
Let b ≥ τ c > τ c g and n ≥ . Assume that Ψ ∈ H m − for aninteger m > n/ . Then there exists a final time T = T ( k Ψ k H m − ) such that problem (4.1) , (4.2) admits a unique mild solution Ψ = ( ψ, v, w, η ) T ∈ C ([0 , T ]; H m − ) , given by Ψ = e t A Ψ + Z t e ( t − r ) A F (Ψ , ∇ Ψ)( r ) d r, where the functional F is defined in (4.5) . HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 13
Proof.
We prove the statement by employing the Banach fixed-point theorem;cf. [19, 20, 24, 38] and [41, Theorem 2.5.4]. To this end, we define the ball B L = { Φ ∈ C ([0 , T ]; H m − ) : k Φ( t ) k H m − ≤ L, ∀ t ∈ [0 , T ] , Φ(0) = Ψ } , equipped with the norm k Φ k B L = sup ≤ t ≤ T k Φ( t ) k H m − . The radius L ≥ k Ψ k H m − of the ball will be conveniently chosen as large enoughbelow. We note that B L is a closed convex subset of C ([0 , T ]; H m − ).For a given Φ = ( ψ φ , v φ , w φ , η φ ) T in B L , we define the mapping T : Φ Ψ byΨ = e t A Ψ + Z t e ( t − r ) A F (Φ , ∇ Φ)( r ) d r. We claim that T is a self-mapping and that it is strictly contractive. We beginby proving that T ( B L ) ⊂ B L . Let Φ ∈ B L . Since m − > n/
2, we have k f ( t ) k H m − = k kv φ ( t ) w φ ( t ) + 2 ∇ ψ φ · ∇ v φ k H m − . k v φ ( t ) k H m − k w φ ( t ) k H m − + k∇ ψ φ ( t ) k H m − k∇ v φ ( t ) k H m − . An application of Young’s inequality immediately yields(5.1) k f ( t ) k H m − . k Φ( t ) k H m − . L , t ∈ [0 , T ] . We are now ready to estimate Ψ( t ): k Ψ( t ) k H m − ≤ k e t A Ψ k H m − + Z t k e ( t − r ) A F (Φ , ∇ Φ) k H m − d r ≤ k Ψ k H m − + Z t k F (Φ , ∇ Φ)( r ) k H m − d r ≤ k Ψ k H m − + Z t k f ( r ) k H m − d r . k Ψ k H m − + T L , where we have employed inequality (5.1) in the last line. Therefore, there exists apositive constant C ⋆ such that k Ψ k B L ≤ C ⋆ ( k Ψ k H m − + T L ) . The final time T can then be chosen small enough and the radius L of the ball B L large enough so that Ψ ∈ B L . Indeed, we first fix L so that C ⋆ k Ψ k H m − ≤ L . With L fixed, we choose the time horizon T > T ≤ C ⋆ L .
This choice of the radius L and final time T yields k Ψ k B L ≤ L, and, therefore, T (Φ) ∈ B L .To prove contractivity, take any Φ = ( ψ φ , v φ , w φ , η φ ) T and Φ ⋆ = ( ψ φ⋆ , v φ⋆ , w φ⋆ , η φ⋆ ) T in B L . We know that kT (Φ) − T (Φ ⋆ ) k H m − ≤ Z t k e ( t − r ) A [ F (Φ , ∇ Φ) − F (Φ ⋆ , ∇ Φ ⋆ )] k H m − d r . Z t k v φ w φ − v φ⋆ w φ⋆ + ∇ ψ φ · ∇ v φ − ∇ ψ φ⋆ · ∇ v φ⋆ k H m − d r. We can further estimate the last term as follows: k v φ w φ − v φ⋆ w φ⋆ + ∇ ψ φ · ∇ v φ − ∇ ψ φ⋆ · ∇ v φ⋆ k H m − . k v φ − v φ⋆ k H m − k w φ k H m − + k v φ⋆ k H m − k w φ − w φ⋆ k H m − + k∇ ( ψ φ − ψ φ⋆ ) k H m − k∇ v φ k H m − + k∇ ψ φ⋆ k H m − k∇ ( v φ − v φ⋆ ) k H m − , recalling that m − > n/
2. Therefore, we have kT (Φ) − T (Φ ⋆ ) k B L . Z t ( k Φ k H m − + k Φ ⋆ k H m − ) k Φ − Φ ⋆ k H m − d r . T L k Φ − Φ ⋆ k B L . By reducing the final time T , we can then guarantee that the mapping T isstrictly contractive. On account of Banach’s fixed-point theorem, a unique solu-tion Ψ = Φ ∈ B L of the problem exists. Uniqueness in C ([0 , T ]; H m − ) followsby assuming that Ψ , Ψ ⋆ ∈ C ([0 , T ]; H m − ) solve the problem and then applyingGronwall’s inequality to k Ψ − Ψ ⋆ k H m − . Z t ( k Ψ k H m − + k Ψ ⋆ k H m − ) k Ψ − Ψ ⋆ k H m − d r. This finishes the proof of Theorem 5.1. (cid:3) Energy estimates in the subcritical case
To prove global solvability, we intend to derive an energy estimate for the so-lution of the nonlinear problem that is uniform in time. Compared to [34], wherelocal nonlinear effects in propagation were neglected, equation (2.4) has a quadraticgradient nonlinearity. This means we have to involve higher-order energies in theestimates. Furthermore, we wish to derive a bound valid for all n ≥ w .According to Theorem 5.1, the nonlinear problem has the solution( ψ, v, w, η ) T ∈ C ([0 , T ]; H m − )for sufficiently short final time. Looking at the third equation in the system, weknow that τ k w t k H m − ≤ k w k H m − + c g k ∆ ψ k H m − + b k ∆ v k H m − + Z ∞ g ( s ) k ∆ η ( s ) k H m − d s + 2 k kvw + ∇ ψ · ∇ v k H m − . HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 15
Similarly to (4.11), we can further estimate the η terms as follows: Z ∞ g ( s ) k ∆ η ( s ) k H m − d s ≤ q c − c g (cid:18)Z ∞ g ( s ) k ∆ η ( s ) k H m − d s (cid:19) / ≤ s c − c g ζ (cid:18)Z ∞ − g ′ ( s ) k ∆ η ( s ) k H m − d s (cid:19) / . Therefore, we have w ∈ C ([0 , T ]; H m − ( R n )) ∩ C ([0 , T ]; H m − ( R n )) .To derive higher-order bounds, we need to work with the space-differentiatedsystem. For simplicity, we introduce the shorthand tilde notation˜ ζ = ∇ κ ζ for ζ ∈ { ψ, v, w, η } , κ ≥ . We then apply the operator ∇ κ ( κ ≥
1) to the system (4.1) to obtain(6.1) ˜ ψ t = ˜ v, ˜ v t = ˜ w,τ ˜ w t = − ˜ w + c g ∆ ˜ ψ + b ∆˜ v + Z ∞ g ( s )∆˜ η ( s ) d s + F ( κ ) ( ψ, v, ∇ ψ, ∇ v ) , ˜ η t = ˜ v − ˜ η s , with the right-hand side nonlinearity given by(6.2) F ( κ ) ( ψ, v, ∇ ψ, ∇ v ) = 2 k [ ∇ κ , v ] w + 2 kv ˜ w + 2 κ [ ∇ κ , ∇ ψ ] · ∇ v + 2 k ∇ ψ · ∇ ˜ v, recalling how the commutator is defined in (3.4).We also introduce two new energy norms at this point. Let n ≥
3. For a giveninteger p ≥
1, we define k Ψ k E p ( t ) = sup ≤ σ ≤ t p X κ =0 E ( κ ) [Ψ]( σ ) , where(6.3) E ( κ ) [Ψ] = k∇ κ +1 ( ψ + τ v ) k L + k∇ κ ( v + τ w ) k L + k∇ κ +1 v k L + k∇ κ +1 η k L , − g ′ + k∇ κ w ( σ ) k L + k ∆ ∇ κ ( ψ + τ v ) k L + k∇ κ +1 ( v + τ w ) k L + k ∆ ∇ κ v k L + k ∆ ∇ κ η k L , − g ′ . With the choice p = m − ≤ σ ≤ t k Ψ( σ ) k H m − .The corresponding dissipation norm is given by k Ψ k D p ( t ) = Z t p X κ =0 D ( κ ) [Ψ] d σ, where we have set(6.4) D ( κ ) [Ψ] = k∇ κ +1 v ( σ ) k L + k∇ κ +1 η ( σ ) k L , − g ′ + k ∆ ∇ κ ( ψ + τ v ) k L + k∇ κ +1 ( v + τ w ) k L + k ∆ ∇ κ v k L + k ∆ ∇ κ η k L , − g ′ + k∇ κ w ( σ ) k L . Our overall goal in the remainder of this section is to prove that the energy k Ψ k E m − ( t ) together with the related quantity k Ψ k D m − ( t ) are uniformly bounded for all time if the energy at initial time is sufficiently small. Following [34, 38], wecan achieve this by first proving that k Ψ k E m − ( t ) + k Ψ k D m − ( t ) . k Ψ k E m − (0) + k Ψ k E m − ( t ) k Ψ k D m − ( t ) . If k Ψ k E m − (0) is small enough, we can then employ Lemma 3.2 to arrive at ourclaim. We prove this inequality in two parts: the first is dedicated to estimatingthe linear and the second to the right-hand side nonlinear terms in the equation.To achieve uniform stability, we assume that we are in the non-critical regime here,where b > τ c .To simplify the notation involving the nonlinear terms, we introduce the right-hand side functionals F ( κ )0 and F ( κ )1 as F ( κ )0 ( ϕ ) = ( F ( κ ) , ϕ ) L , F ( κ )1 ( ϕ ) = ( ∇ F ( κ ) , ∇ ϕ ) L , where κ ∈ { , , . . . , m − } , and ϕ stands for different test functions that will beused below.We set aside estimates of the nonlinear terms for a moment and focus on esti-mating the linear terms in the equation. Proposition 6.1.
Let b > τ c > τ c g . Then for each κ ∈ { , , . . . , m − } , it holds (6.5) sup ≤ σ ≤ t E ( κ ) [Ψ]( σ ) + Z t D ( κ ) [Ψ]( σ ) d σ . E ( κ ) [Ψ ] + Z t n |F ( κ )0 ( ∇ κ ( v + τ w )) | + |F ( κ )1 ( ∇ κ ( v + τ w )) | + |F ( κ )0 ( ∇ κ w ) | + |F ( κ )1 ( ∇ κ ( ψ + τ v )) | + |F ( κ )1 ( τ ∇ κ v ) | o d σ. Proof.
The proof follows by performing the energy analysis of [34] on the space-differentiated system (6.1), so we provide only an outline of the arguments hereand refer to [34] for details. Similarly to our previous reasoning, we first need tointroduce energies that are tailored to our particular problem. The first one is givenby E ( κ )1 ( t ) = 12 (cid:20) c g k∇ κ +1 ( ψ + τ v ) k L + τ ( b − τ c g ) k∇ κ +1 v k L + k∇ κ ( v + τ w ) k L + τ k∇ κ +1 η k L , − g ′ + k∇ κ +1 η k L ,g +2 τ Z R n Z ∞ g ( s ) ∇ κ +1 η ( s ) · ∇ κ +1 v d s d x (cid:21) at time t ≥
0. Applying the energy estimates in [34, Proposition 4.1] to the systemsatisfied by ( ∇ κ ψ, ∇ κ v, ∇ κ w, ∇ κ η ) yields the following dissipativity relation:(6.6) dd t E ( κ )1 ( t ) + ( b − τ c ) k∇ κ +1 v ( t ) k L + 12 k∇ κ +1 η k L , − g ′ ≤ |F ( κ )0 ( ∇ κ ( v + τ w )) | for all t ≥
0. Equipped with the arguments presented in the proof of Proposition 4.1,it is straightforward to check that this problem-specific energy is equivalent to E ( κ )1 [Ψ] = k∇ κ +1 ( ψ + τ v ) k L + k∇ κ ( v + τ w ) k L + k∇ κ +1 v k L + k∇ κ +1 η k L , − g ′ . HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 17
Secondly, we introduce the energy of order κ + 2 by E ( κ )2 ( t ) = 12 (cid:20) c g k ∆( ∇ κ ( ψ + τ v )) k L + τ ( b − τ c g ) k ∆ ∇ κ v k L + k∇ κ +1 ( v + τ w ) k L + τ k ∆ ∇ κ η k L , − g ′ + k ∆ ∇ κ η k L ,g +2 τ Z R n Z ∞ g ( s )∆ ∇ κ v ∆ ∇ κ η ( s ) d s d x (cid:21) . This energy is clearly equivalent to E ( κ +1)1 and thus to E ( κ )2 [Ψ] = k ∆ ∇ κ ( ψ + τ v ) k L + k∇ κ +1 ( v + τ w ) k L + k ∆ ∇ κ v k L + k ∆ ∇ κ η k L , − g ′ . It also satisfies a dissipative relation in the folowing form:(6.7) dd t E ( κ )2 ( t ) + (cid:0) b − τ c (cid:1) k ∆ ∇ κ v k L + 12 k ∆ ∇ κ η k L , − g ′ ≤ |F ( κ )1 ( ∇ κ ( v + τ w )) | ;see [34, Proposition 4.2] for the proof in the case κ = 0. To capture dissipation ofterms k∇ κ +1 ( ψ + τ v ) k L and k∇ κ +1 ( v + τ w ) k L , we introduce the functionals F ( κ )1 ( t ) = Z R n ∇ κ +1 ( ψ + τ v ) · ∇ κ +1 ( v + τ w ) d x, and F ( κ )2 ( t ) = − τ Z R n ∇ κ +1 v · ∇ κ +1 ( v + τ w ) d x, for all t ≥
0, where κ ∈ { , , . . . , m − } . They satisfy suitable dissipative relations.It can be shown thatdd t F ( κ )1 ( t ) + ( c g − ǫ − ( c − c g ) ǫ ) k ∆ ∇ κ ( ψ + τ v ) k L ≤ k∇ κ +1 ( v + τ w ) k L + C ( ǫ ) k ∆ ∇ κ v k L + C ( ǫ ) k ∆ ∇ κ η k L ,g + |F ( κ )1 ( ∇ κ ( ψ + τ v )) | . for any positive ǫ , ǫ >
0. Furthermore, for any ǫ , ǫ >
0, we havedd t F ( κ )2 ( t ) + (1 − ǫ ) k∇ κ +1 ( v + τ w ) k L ≤ ǫ k ∆ ∇ κ ( ψ + τ v ) k L + C ( ǫ , ǫ )( k ∆ ∇ κ v k L + k∇ κ +1 v k L )+ 12 k∇ κ +1 η k L ,g + |F ( κ )1 ( τ ∇ κ v ) | ;see Lemmas 4.3 and 4.4 in [34] for the case κ = 0, the arguments can easily beadapted to include κ ∈ { , . . . , m − } . Testing the space-differentiated third equa-tion in the system by ∇ κ w yields12 dd t k∇ κ w ( t ) k L + 12 k∇ κ w k L . k ∆ ∇ κ ( ψ + τ v ) k L + k ∆ ∇ κ v k L + k ∆ ∇ κ η k L ,g + |F ( κ )0 ( ∇ κ w ) | . We are ready to define the Lyapunov functional L ( κ ) of order κ as(6.8) L ( κ ) ( t ) = L ( E ( κ )1 ( t ) + E ( κ )2 ( t ) + ετ k∇ κ w ( t ) k L ) + F ( κ )1 ( t ) + L F ( κ )2 ( t ) , at time t ≥
0. There exist a constant L > ε > κ ≥ t L ( κ ) ( t ) + k∇ κ +1 v ( t ) k L + k∇ κ +1 η k L , − g ′ + E ( κ )2 [Ψ]( t ) + k∇ κ w ( t ) k L . |F ( κ )0 ( ∇ κ ( v + τ w )) | + |F ( κ )1 ( ∇ κ ( v + τ w )) | + |F ( κ )0 ( ∇ κ w ) | + |F ( κ )1 ( ∇ κ ( ψ + τ v )) | + |F ( κ )1 ( τ ∇ κ v ) | , for all t ∈ [0 , T ]. We note that having b > τ c in (6.6) and (6.7) is essential inobtaining dissipativity of this Lyapunov functional.By integrating estimate (6.9) over the time interval (0 , σ ) for σ ∈ (0 , t ) and thentaking the supremum over time, we obtain (6.5). (cid:3) The remaining challenge in the uniform energy analysis is to control the terms F ( κ )0 and F ( κ )1 in estimate (6.5). To formulate the next results, we introduce herethe functional(6.10) Λ[Ψ]( t ) = sup ≤ σ ≤ t (cid:16) k v ( σ ) k W , ∞ + k w ( σ ) k L ∞ + k∇ ψ ( σ ) k L ∞ + k∇ ψ ( σ ) k H n − + k∇ ψ ( σ ) k H n − + k v ( σ ) k H n − + k∇ v ( σ ) k H n − + k w ( σ ) k H n − (cid:17) . We prove the estimates for the cases κ = 0 and κ ∈ { , . . . , m − } separately. Theorem 6.1.
Let b > τ c > τ c g and n ≥ . Then it holds that (6.11) k Ψ k E ( t ) + k Ψ k D ( t ) . k Ψ k E (0) + Λ[Ψ]( t ) k Ψ k D ( t ) . Proof.
To prove the statement, we should estimate the integral terms on the right-hand side of (6.5). Using H¨older’s inequality yields(6.12) (cid:12)(cid:12)(cid:12) F (0)0 ( v + τ w ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k Z R n ( vw + ∇ ψ · ∇ v ) ( v + τ w ) d x (cid:12)(cid:12)(cid:12)(cid:12) . Z R n | vw ( v + τ w ) | d x + Z R n |∇ ψ · ∇ v ( v + τ w ) | d x. To estimate the terms on the right, we begin with noting that Z R n (cid:12)(cid:12) wv (cid:12)(cid:12) d x . k w k L k v k L n k v k L nn − . k w k L k v k H n − k∇ v k L , where we have used the estimate (3.2) for tri-linear terms, the endpoint Sobolevembedding (3.1), and the embedding (3.3). Consequently, we obtain(6.13) Z t Z R n (cid:12)(cid:12) wv (cid:12)(cid:12) d x d σ . sup ≤ σ ≤ t k v ( σ ) k H n − Z t ( k w ( σ ) k L + k∇ v ( σ ) k L ) d σ . sup ≤ σ ≤ t k v ( σ ) k H n − k Ψ k D ( t ) . Similarly, we have (cid:12)(cid:12)(cid:12)(cid:12)Z R n vw d x (cid:12)(cid:12)(cid:12)(cid:12) . k w k L k v k L n k w k L nn − . k w k L k v k H n − k∇ w k L , from which it follows that(6.14) Z t Z R n (cid:12)(cid:12) vw d x (cid:12)(cid:12) d σ . sup ≤ σ ≤ t k v ( σ ) k H n − k Ψ k D ( t ) . HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 19
By adding estimates (6.13) and (6.14), we find that Z t Z R n | vw ( v + τ w ) | d x d σ . sup ≤ σ ≤ t k v ( σ ) k H n − k Ψ k D ( t ) . For the remaining terms on the right in (6.12), we have by H¨older’s inequality(6.15) Z t Z R n |∇ ψ · ∇ v ( τ w ) | d x d σ . sup ≤ σ ≤ t k∇ ψ ( σ ) k L ∞ k Ψ k D ( t ) We also note that Z R n | v ∇ ψ · ∇ v | d x . k∇ v k L k∇ ψ k L n k v k L nn − . k∇ v k L k∇ ψ k H n − k∇ v k L , where we have again used the estimate (3.2), the endpoint Sobolev embedding (3.1),and the embedding (3.3). This immediately yields(6.16) Z t Z R n | v ∇ ψ · ∇ v | d x d s . sup ≤ σ ≤ t k∇ ψ ( σ ) k H n − k Ψ k D ( t ) . From estimates (6.15) and (6.16), we conclude that Z t Z R n |∇ ψ · ∇ v ( v + τ w ) | d x d σ . sup ≤ σ ≤ t (cid:16) k∇ ψ ( σ ) k L ∞ + k∇ ψ ( σ ) k H n − (cid:17) k Ψ k D ( t ) . Altogether, we have proven that(6.17) Z t |F (0)0 ( v + τ w )( σ ) | d σ . sup ≤ σ ≤ t (cid:16) k∇ ψ ( σ ) k L ∞ + k∇ ψ ( σ ) k H n − + k v ( σ ) k H n − (cid:17) k Ψ k D ( t ) . We can estimate the term R t F (0)0 ( w )( σ ) d σ in (6.5) similarly, so we move on toestimating the remaining three F (0)1 terms. We can rewrite F (0)1 ( v + τ w ) as F (0)1 ( v + τ w )= 2 k Z R n ∇ (cid:18) τ v ( v + τ w − v ) + 2 ∇ ( ψ + τ v − τ v ) · ∇ v (cid:19) · ∇ ( v + τ w ) d x = 2 k Z R n (cid:18) τ v ∇ ( v + τ w ) + 1 τ ∇ v ( v + τ w ) − ∇ | v | (cid:19) · ∇ ( v + τ w ) d x + Z R n (2 H ( ψ + τ v ) ∇ v + 2 H ( v ) ∇ ( ψ + τ v ) − τ H ( v ) ∇ v ) · ∇ ( v + τ w ) d x, where we have introduced the following notation for the Hessian matrix of f : H ( f ) = ( ∂ x i ∂ x j f ) , ≤ i,j ≤ n . By using the identity k H ( f ) k L = k ∆ f k L , together with H¨older’s inequality, we infer |F (0)1 ( v + τ w ) | . k∇ v k L ( k∇ ( v + τ w ) k H n − + k∇ v k H n − ) k∇ ( v + τ w ) k L + k∇ ( v + τ w ) k L k∇ v k H n − + k∇ v k L ∞ ( k ∆( ψ + τ v ) k L + k ∆ v k L ) k∇ ( v + τ w ) k L + k∇ ( ψ + τ v ) k L ∞ k ∆ v k L k∇ ( v + τ w ) k L . Integrating the above inequality from 0 to t , yields Z t |F (0)1 ( v + τ w )( σ ) | d σ . sup ≤ σ ≤ t ( k∇ v ( σ ) k H n − + k∇ v ( σ ) k L ∞ + k∇ ( v + τ w )( σ ) k H n − + k∇ ( ψ + τ v )( σ ) k L ∞ ) k Ψ k D ( t ) . Similarly, we have the following bound: Z t |F (0)1 ( ψ + τ v )( σ ) | d σ . sup ≤ σ ≤ t (cid:0) k w ( σ ) k H n − + k∇ ψ ( σ ) k L ∞ (cid:1) k Ψ k D ( t ) , as well as(6.18) Z t |F (0)1 ( τ v )( σ ) | d σ . sup ≤ σ ≤ t (cid:0) k w ( σ ) k H n − + k∇ ψ ( σ ) k L ∞ (cid:1) k Ψ k D ( t ) . By plugging estimates (6.17)–(6.18) into (6.5), we obtain the desired bound. (cid:3)
We next prove an analogous result when κ ∈ { , . . . , m − } . Theorem 6.2.
Let b > τ c > τ c g and n ≥ . For any t ≥ and for any κ ∈{ , . . . , m − } , it holds that (6.19) sup ≤ σ ≤ t E ( κ ) [Ψ]( σ ) + Z t D ( κ ) [Ψ]( σ ) d σ . E ( κ ) [Ψ](0) + Λ[Ψ]( t ) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ, where the function Λ[Ψ] is defined in (6.10) . The functionals E ( κ ) [Ψ] and D ( κ ) [Ψ] are defined in (6.3) and (6.4) , respectively.Proof. As in the previous proof, the crucial part is to get appropriate estimates onthe five integral terms on the right-hand side of (6.5). Here we have to rely on thecommutator bounds. We begin by noting that(6.20) |F ( κ )0 ( ∇ κ ( v + τ w ))( σ ) | = |F ( κ )0 ((˜ v + τ ˜ w ))( σ ) | . Z R n | [ ∇ κ , v ] w || (˜ v + τ ˜ w ) | d x + Z R n | v ˜ w || (˜ v + τ ˜ w ) | d x + Z R n | [ ∇ κ , ∇ ψ ] ∇ v || (˜ v + τ ˜ w ) | d x + Z R n |∇ ψ · ∇ ˜ v || (˜ v + τ ˜ w ) | d x. Starting from the last term on the right, we have Z R n |∇ ψ · ∇ ˜ v || (˜ v + τ ˜ w ) | d x ≤ Z R n | ˜ v ∇ ψ · ∇ ˜ v | d x + Z R n | τ ∇ ψ · ∇ ˜ v ˜ w | d x. We can bound the second term on the right as follows: Z R n | τ ∇ ψ · ∇ ˜ v ˜ w | d x . k∇ ψ k L ∞ k∇ ˜ v k L k ˜ w k L , which by H¨older’s inequality yields Z t Z R n | τ ∇ ψ · ∇ ˜ v ˜ w | d x d σ . sup ≤ σ ≤ t k∇ ψ ( σ ) k L ∞ Z t D ( κ ) [Ψ]( σ ) d σ. HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 21
We next have Z R n | ˜ v ∇ ψ · ∇ ˜ v | d x . k ˜ v k L nn − k∇ ψ k L n k∇ ˜ v k L . k∇ ˜ v k L k∇ ψ k L n k∇ ˜ v k L , which leads to Z t Z R n | ˜ v ∇ ψ · ∇ ˜ v | d x d σ . sup ≤ σ ≤ t k∇ ψ ( σ ) k H n − n Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. We deduce from the derived bounds that Z t Z R n |∇ ψ · ∇ ˜ v || (˜ v + τ ˜ w ) | d x d σ . sup ≤ σ ≤ t (cid:0) k∇ ψ ( σ ) k L ∞ + k∇ ψ ( σ ) k H n − (cid:1) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. We next want to estimate the second term on the right in (6.20):(6.21) Z R n | v ˜ w || (˜ v + τ ˜ w ) | d x = Z R n | v ˜ w || (˜ v + τ ˜ w ) | d x ≤ Z R n | v ˜ w ˜ v | d x + τ Z R n | v ˜ w | d x. We further have τ Z R n | v ˜ w | d x . k ˜ w k L nn − k v k L n k ˜ w k L . k∇ ˜ w k L k v k L n k ˜ w k L , which yields Z t τ Z R n | v ˜ w | d x d σ . sup ≤ σ ≤ t k v ( σ ) k H n − Z t D ( κ ) [Ψ]( σ ) d σ. We can derive the same bound for the first term on the right in (6.21) after inte-gration in time. Consequently, we have the following estimate: Z t Z R n | v ˜ w || (˜ v + τ ˜ w ) | d x d σ . sup ≤ σ ≤ t k v ( σ ) k H n − Z t D ( κ ) [Ψ]( σ ) d σ. We next estimate the first term on the right-hand side in (6.20): Z R n | [ ∇ κ , v ] w || (˜ v + τ ˜ w ) | d x . k [ ∇ κ , v ] w k L nn +2 k (˜ v + τ ˜ w ) k L nn − . k [ ∇ κ , v ] w k L nn +2 ( k ˜ v + τ ˜ w k L + k∇ (˜ v + τ ˜ w ) k L ) . Thus, we have by applying the Sobolev embedding (3.3) k [ ∇ κ , v ] w k L nn +2 . ( k∇ v k L n k∇ κ − w k L + k w k L n k∇ κ v k L ) . ( k∇ v k H n − + k w k H n − )( k∇ κ − w k L + k∇ κ v k L ) . Therefore, we obtain Z t Z R n | [ ∇ κ , v ] w || (˜ v + τ ˜ w ) | d x d σ . sup ≤ σ ≤ t (cid:16) k∇ v ( σ ) k H n − + k w ( σ ) k H n − (cid:17) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. It remains to estimate the third term on the right in (6.20): Z R n | [ ∇ κ , ∇ ψ ] ∇ v || (˜ v + τ ˜ w ) | d x . k [ ∇ κ , ∇ ψ ] ∇ v k L nn +2 k (˜ v + τ ˜ w ) k L nn − . k [ ∇ κ , ∇ ψ ] ∇ v k L nn +2 ( k ˜ v + τ ˜ w k L + k∇ (˜ v + τ ˜ w ) k L ) . Now, by applying once more the embedding H n − ( R n ) ֒ → L n ( R n ), we have k [ ∇ κ , ∇ ψ ] ∇ v k L nn +2 . ( k∇ ψ k L n k∇ κ v k L + k∇ v k L n k∇ κ +1 ψ k L ) . ( k∇ ψ k H n − + k∇ v k H n − )( k∇ κ v k L + k∇ κ +1 ψ k L ) . Hence, we obtain from the above two estimates Z t Z R n | [ ∇ κ , ∇ ψ ] ∇ v || (˜ v + τ ˜ w ) | d x d σ . sup ≤ σ ≤ t (cid:16) k∇ ψ ( σ ) k H n − + k∇ v ( σ ) k H n − (cid:17) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. Altogether, we have proven the following estimate for F ( κ )0 ( ∇ κ ( v + τ w )): Z t |F ( κ )0 ( ∇ κ ( v + τ w ))( σ ) | d σ . Λ[Ψ]( t ) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ, with the function Λ[Ψ] defined in (6.10).Next we claim that(6.22) Z t ( |F ( κ )1 ( ∇ κ ( v + τ w ))( σ ) | + |F ( κ )1 ( τ ∇ κ v )( σ ) | ) d σ . Λ( t ) Z t D ( κ ) [Ψ]( σ ) d σ. We refer to [38] for the proof when n = 3. We find by the Cauchy–Schwarz inequalitythat(6.23) |F ( κ )1 ( ∇ κ ( v + τ w )) | ( σ ) + |F ( κ )1 ( τ ∇ κ v ) |≤ k∇ F ( κ ) k L ( k∇ (˜ v + τ ˜ w ) k L + k∇ ˜ v k L ) . Keeping in mind how the functional F ( κ ) is defined in (6.2), we obtain(6.24) k∇ F ( κ ) k L . k w k L ∞ k∇ κ +1 v k L + k v k L ∞ k∇ κ +1 w k L + k∇ ψ k L ∞ k∇ κ +2 v k L + k∇ v k L ∞ k∇ κ +2 ψ k L . where we have used inequality (3.5). We then estimate k∇ ψ k L ∞ k∇ κ +2 v k L + k∇ v k L ∞ k∇ κ +2 ψ k L . ( k∇ ψ k L ∞ k ∆ ∇ κ v k L + k∇ v k L ∞ k ∆ ∇ κ ψ k L ) . k∇ ψ k L ∞ k ∆ ∇ κ v k L + k∇ v k L ∞ ( k ∆ ∇ κ ( ψ + τ v ) k L + k ∆ ∇ κ v k L ) . Inserting the above estimates into (6.24) yields k∇ F ( κ ) k L . Λ[Ψ]( t )( k∇ ˜ v k L + k∇ ˜ w k L + k ∆˜ v k L + k ∆( ˜ ψ + τ ˜ v ) k L ) . The above bound taken together with estimate (6.23) implies (6.22). We can esti-mate |F ( κ )0 ( ∇ κ w ) | analogously to arrive at Z t |F ( κ )0 ( ∇ κ w )( σ ) | d σ . Λ[Ψ]( t ) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 23
Finally, we provide an estimate of |F ( κ )1 ( ∇ κ ( ψ + τ v )) | . Observe that |F ( κ )1 ( ∇ κ ( ψ + τ v )) | = |F ( κ )0 (∆ ∇ κ ( ψ + τ v )) | . Hence we have |F ( κ )0 (∆ ∇ κ ( ψ + τ v )) | ≤ k F ( κ ) k L k ∆( ˜ ψ + τ ˜ v ) k L . Again by keeping in mind how F ( κ ) is defined in (6.2), we have(6.25) k F ( κ ) k L . k [ ∇ κ , v ] w k L + k v k L ∞ k ˜ w k L + k [ ∇ κ , ∇ ψ ] ∇ v k L + k∇ ψ k L ∞ k∇ ˜ v k L . By exploiting the commutator estimate from Lemma 3.1, we find k [ ∇ κ , v ] w k L . ( k∇ v k L ∞ k∇ κ − w k L + k w k L ∞ k∇ κ v k L ) . Similarly, k [ ∇ κ , ∇ ψ ] ∇ v k L . ( k∇ v k L ∞ k∇ κ +1 ψ k L + k∇ ψ k H n − k∇ κ v k H ) . k∇ v k L ∞ (cid:0) k∇ κ +1 ( ψ + τ v ) k L + k∇ κ +1 v k L (cid:1) + k∇ ψ k H n − ( k∇ κ v k L + k∇ κ +1 v k L ) . Plugging the last two estimates into (6.25) results in k F ( κ ) k L . Λ[Ψ]( t ) (cid:0) k∇ κ − w k L + k∇ κ +1 v k L + k∇ κ − ∇ v k L + k∇ κ − ∆ ( ψ + τ v ) k L (cid:1) . Altogether, we have Z t |F ( κ )1 ( ∇ κ ( ψ + τ v )) | d σ . Λ[Ψ]( t ) Z t ( D ( κ − [Ψ]( σ ) + D ( κ ) [Ψ]( σ )) d σ. This completes the proof. (cid:3)
Having estimated the nonlinear terms, we are now ready to prove the final energybound.
Theorem 6.3.
Let b > τ c > τ c g . Let n ≥ and m > n/ be an integer. Thenthe following estimate holds: (6.26) k Ψ k E m − ( t ) + k Ψ k D m − ( t ) . k Ψ k E m − (0) + k Ψ k E m − ( t ) k Ψ k D m − ( t ) . Proof.
By summing up (6.19) over κ = 1 , . . . , m −
2, and adding the result toestimate (6.11) ( κ = 0), we find(6.27) k Ψ k E m − ( t ) + k Ψ k D m − ( t ) . k Ψ k E m − (0) +(Λ[Ψ]( t )+ k Ψ k E m − ( t ) ) k Ψ k D m − ( t ) . It remains to estimate Λ[Ψ]( t ). Since m > n/ ∇ ψ, ∇ v ∈ H m − ( R n ) ֒ → L ∞ ( R n ) , ∇ ψ, ∇ ψ, ∇ v, ∇ w ∈ H m − ( R n ) ֒ → H n − ( R n ) ,v, w ∈ H m − ( R n ) ֒ → H n − ( R n ) , to infer that Λ[Ψ]( t ) . k Ψ k E m − ( t ) , t ∈ [0 , T ] . By plugging the above bound into (6.27), we conclude that (6.26) holds true. (cid:3) Global solvability in the subcritical case
Equipped with the uniform bound (6.26), we can now prove global solvability ofthe JMGT equation with memory in R n , where n ≥ Theorem 7.1.
Let b > τ c > τ c g and n ≥ . Assume that Ψ ∈ H m − for aninteger m > n/ . Then there exists a positive constant δ, such that if k Ψ k E m − (0) ≤ δ, then problem (4.1) , (4.2) has a global solution Ψ ∈ { Ψ = ( ψ, v, w, η ) T : Ψ ∈ C ([0 , ∞ ); H m ) } . Proof.
Let
T > ||| Ψ ||| (0 ,t ) = k Ψ k E m − ( t ) + k Ψ k D m − ( t ) is uniformly bounded for all time provided that the initial energy is sufficientlysmall. We have k Ψ k B L (0 ,t ) ≤ k Ψ k E m − ( t ) + k Ψ k D m − ( t ) = ||| Ψ ||| (0 ,t ) , where k Ψ k B L (0 ,t ) = sup ≤ σ ≤ t k Φ k H s ≡ k Φ k E m − ( t ) Thanks to the previous section, we have the energy bound k Ψ k E m − ( t ) + k Ψ k D m − ( t ) . k Ψ k E m − (0) + k Ψ k E m − ( t ) k Ψ k D m − ( t ) , t ∈ [0 , T ] . Therefore, for all t ∈ [0 , T ], we have ||| Ψ ||| (0 ,t ) ≤ k Ψ k E m − (0) + C ||| Ψ ||| / ,t ) , (7.1)Thanks to Lemma 3.2, this means that there exists a positive constant C , indepen-dent of t , such that ||| Ψ ||| (0 ,t ) ≤ C. This uniform bound implies that the local solution can be extended to T = ∞ . (cid:3) Accordingly, the JMGT equation in hereditary media with quadratic gradient non-linearity and initial conditions Ψ ∈ H m − admits a unique solution ψ such that ∇ ψ ∈ C ([0 , + ∞ ); H m − ( R n )) ∩ C ([0 , + ∞ ); H m − ( R n )) ,ψ t ∈ C ([0 , + ∞ ); H m ( R n )) ∩ C ([0 , + ∞ ); H m − ( R )) ,ψ tt ∈ C ([0 , + ∞ ); H m − ( R n )) ∩ C ([0 , + ∞ ); H m − ( R n )) , where m > n/ b > τ c >τ c g holds. In particular, in a three-dimensional setting, we have m ≥ HE JMGT WAVE EQUATION IN HEREDITARY FLUIDS 25
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Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJNijmegen, The Netherlands
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