Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane
Ying-Chieh Lin, C. H. Arthur Cheng, John M. Hong, Jiahong Wu, Juan-Ming Yuan
aa r X i v : . [ m a t h . A P ] M a y WELL-POSEDNESS OF THE TWO-DIMENSIONAL GENERALIZEDBENJAMIN-BONA-MAHONY EQUATION ON THE UPPER HALFPLANE
YING-CHIEH LIN, C. H. ARTHUR CHENG, JOHN M. HONG, JIAHONG WU,AND JUAN-MING YUAN
Abstract.
This paper focuses on the two-dimensional Benjamin-Bona-Mahonyand Benjamin-Bona-Mahony-Burgers equations with a general flux function. Theaim is at the global (in time) well-posedness of the initial-and boundary-valueproblem for these equations defined in the upper half-plane. Under suitable growthconditions on the flux function, we are able to establish the global well-posednessin a Sobolev class. When the initial- and boundary-data become more regular,the corresponding solutions are shown to be classical. In addition, the continuousdependence on the data is also obtained. Introduction
This paper is concerned with the two-dimensional (2D) Benjamin-Bona-Mahony-Burgers equation of the form u t + div ( φ ( u )) = ν ∆ u + ν ∆ u t in Ω × (0 , T ) , (1.1a) u = g on Ω × { t = 0 } , (1.1b) u = h on ∂ Ω × (0 , T ) , (1.1c)where Ω ⊆ R denotes a smooth domain, u = u ( x, t ) is a scalar function, φ ( u ) is avector-valued flux function, and ν ≥ ν > u t + u x + uu x − u xxt = 0 , (1.2)which governs the unidirectional propagation of 1D long waves with small amplitudes.Therefore (1.1a) with ν = 0 is sometimes called the generalized Benjamin-Bona-Mahony (GBBM) equation while (1.1a) with ν > u t + div( φ ( u )) = 0 . (1.3) Mathematics Subject Classification.
Key words and phrases.
Generalized Benjamin-Bona-Mahony equation, Benjamin-Bona-Mahony-Burgers equation, Buckley-Leverett equation, initial-boundary value problem, global well-posedness.
In addition, (1.1a) has also been derived to model the two-phase fluid flow in a porousmedium, as in the oil recovery. In fact, (1.1a) is a special case of the well-knownBuckley-Leverett equation u t + div( φ ( u )) = − div { H ( u ) ∇ ( J ( u ) − τ u t ) } , (1.4)where u denotes the saturation of water, the functions φ , H and J are related to thecapillary pressure and the permeability of water and oil [14]. (1.1a) follows from (1.4)by linearizing the static capillary pressure J ( u ) and H around a constant state.Attention here will be focused on the case when Ω = R , the upper half-plane.The aim is at the global well-posedness of (1.1) with inhomogeneous boundary data,namely h
0. One motivation behind this study is to rigorously validate the labo-ratory experiments involving water waves generated by a wavemaker mounted at theend of a water channel. We are able to prove the global existence and uniqueness ofthe mild and classical solutions to (1.1). In addition, a continuous dependence resultis also obtained. Our main theorems can be stated as follows.
Theorem 1.1 (Existence and uniqueness) . Let ν ≥ , ν > , and Ω = R . Let T > . Suppose that ( g, h ) ∈ H ( R ) × C ([0 , T ]; H ( R )) , and the flux φ ∈ C ( R , R ) satisfies the conditions φ (0) = 0 and k φ ′′ k L ∞ ( R ) ≤ C. (1.5) Then (1.1) admits a unique mild solution u ∈ C ([0 , T ]; H ( R )) . If we further assumethat ( g, h ) ∈ C ,α loc ( R ) × C ([0 , T ]; C ,α loc ( R )) for some < α < , then the mild solutionis in fact a classical solution. We remark that the condition φ (0) = 0 in (1.5) can be removed. In the caseof φ (0) = 0, we define the new function e φ ( s ) = φ ( s ) − φ (0) for all s ∈ R , then e φ (0) = 0 and div( e φ ( v )) = div( φ ( v )). We can consider the new equation by replacingthe function φ with e φ in (1.1a). Theorem 1.2 (Continuous dependence on data) . Let ν ≥ , ν > , and Ω = R .Suppose that φ ∈ C ( R , R ) satisfies the condition (1.5) . Then the mild solutionobtained in Theorem depends continuously on the initial datum g and boundarydatum h . If we further assume that φ ∈ C ( R , R ) satisfies k φ ′′′ k L ∞ ( R ) ≤ C , (1.6) then the same result also holds for the classical solution.
D GBBM EQUATION 3
It is worth remarking that Theorems 1.1 and 1.2 hold with either ν = 0 or with ν > − ∆ in 2D is much more singular than the 1D case; and second,the inhomogeneous boundary data prevents us from obtaining a time-independent H upper bound, which very much simplifies the process of global-in-time estimates. Toovercome the difficulties, we introduce a new function that assumes the homogeneousboundary data and rewrite the equation in an integral form through the Green func-tion of the elliptic operator. In addition, we use the bootstrapping technique to obtainthe classical solution of (1.1) instead of looking for the solution in classical spacesdirectly.The rest of this paper is divided into six sections. The first five sections deal withthe case when ν = 0 while the last section explains why the results for ν = 0 canbe extended to the case when ν >
0. Section 2 introduces a new function thatassumes homogenous boundary data and converts (1.1) into an integral formulationin terms of this new function. Section 3 presents preliminary regularity estimates forthe operator ( I − ∆) − in the Sobolev space H and in H¨older spaces. Section 4 provesthat (1.1) has a unique local (in time) classical solution. We make use of the integralrepresentation (2.5). Section 5 establishes the global existence and uniqueness of the Y.-C. LIN, C. H. A. CHENG, J. M. HONG, J. WU, AND J.-M. YUAN local solution obtained in Section 4 by showing global bounds for the solution in H and in H¨older spaces. Section 6 contains the continuous dependence results. Thecontinuous dependence of the solution on the initial data and the boundary data isproven in two functional settings and the proof is lengthy. As aforementioned, Section7 is devoted to the case when ν > An alternative formulation
In this section we set ν = 0. The case when ν > v ( x, t ) = u ( x, t ) − h ( x , t ) e − x , which satisfies(I − ∆) v t + div (cid:0) φ ( v + he − x ) (cid:1) = h x x t e − x in R × (0 , T ) , (2.1a) v = e g on R × { t = 0 } , (2.1b) v = 0 on ∂ R × (0 , T ) . (2.1c)where e g ( x ) = g ( x ) − h ( x , e − x . (2.2)Denoting (I − ∆) − f as the unique solution to the elliptic equation(I − ∆) u = f in Ω , (2.3a) u = 0 on ∂ Ω , (2.3b)we can formally write the solution v of (2.1) via the integral representation v ( x, t ) = e g ( x ) + (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1) − Z t (I − ∆) − (cid:8) div (cid:0) φ ( v + he − x ) (cid:1)(cid:9) dτ . (2.4)For short, we rewrite (2.4) as the form v = A v = e g + B h + C v, (2.5)where, for x ∈ R and t ≥ B h ( x, t ) := (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1) , C v ( x, t ) := − Z t (I − ∆) − (cid:8) div (cid:0) φ ( v + he − x ) (cid:1)(cid:9) dτ. D GBBM EQUATION 5 Preliminary results
This section specifies the functional spaces and provides two preliminary estimateson the solutions to the elliptic equation (2.3). In the rest of this paper, we write C k ([0 , T ]; H ℓ ( R )) for the space n u : [0 , T ] → H ℓ ( R ) (cid:12)(cid:12)(cid:12) lim t → t k X j =0 (cid:13)(cid:13)(cid:13) ∂ k u∂t k ( t ) − ∂ k u∂t k ( t ) (cid:13)(cid:13)(cid:13) H ℓ ( R ) = 0 ∀ t ∈ [0 , T ] o equipped with norm k u k C kt H ℓx := max t ∈ [0 ,T ] k X j =0 (cid:13)(cid:13)(cid:13) ∂ k u∂t k ( t ) (cid:13)(cid:13)(cid:13) H ℓ ( R ) . The spaces with the particular indices k = 0 , ℓ = 1 , k = 0, we omit the super-index 0, that is, C ([0 , T ]; H ℓ ( R )) ≡ C ([0 , T ]; H ℓ ( R )) and k · k C t H ℓx ≡ k · k C t H ℓx . We will also needthe space C ([0 , T ]; L p ( R )) ≡ n u : [0 , T ] → L p ( R ) (cid:12)(cid:12)(cid:12) lim t → t k u ( t ) − u ( t ) k L p ( R ) = 0 o equipped with norm k u k C t L px = max t ∈ [0 ,T ] k u ( t ) k L p ( R ) . Similar notation is used to define the space of the boundary data which is only definedon the real line R . We introduce C ([0 , T ]; H ( R )) ≡ n h : [0 , T ] → H ( R ) (cid:12)(cid:12)(cid:12) lim t → t k h ( t ) − h ( t ) k H ( R ) = 0 o equipped with norm k h k C t H x = max t ∈ [0 ,T ] k h ( t ) k H ( R ) . To study the classical solutions, we let C k,α (Ω) denote the space of k -times clas-sically differentiable functions whose k -th derivatives are H¨older continuous with ex-ponent α . The norm on C k,α (Ω) is given by k u k C k,α (Ω) = k X j =0 sup x ∈ Ω | D j u ( x ) | + sup x,y ∈ Ω | D k u ( x ) − D k u ( y ) || x − y | α , where D j u denotes the j -th classical derivative of u .To deal with the integral representation (2.4), we need some crucial estimates onthe operator (I − ∆) − . In particular, the bounds in the following propositions willbe employed in the subsequent sections. Y.-C. LIN, C. H. A. CHENG, J. M. HONG, J. WU, AND J.-M. YUAN
Proposition 3.1.
Assume f ∈ L (Ω) . Then the Dirichlet problem (2.3) admits aunique solution u ∈ H (Ω) . Furthermore, k u k H (Ω) ≤ C k f k L (Ω) . If f is instead in a H¨older space, then we have the following H¨older’s estimates forthe solution of (2.3). Proposition 3.2.
Assume that Ω ⊂ R is a smooth domain. Assume that f is in C ,α loc (Ω) ∩ L (Ω) for some < α < . Then the solution u of the Dirichlet problem (2.3) lies in C ,α loc (Ω) ∩ H (Ω) . Furthermore, for any compact subsets Ω and Ω of Ω with Ω ⊂⊂ Ω , k u k C ,α (Ω ) ≤ C ( k f k C ,α (Ω ) + k f k L (Ω ) ) , where C > depends only on the distance between Ω and ∂ Ω .Proof. By Proposition 3.1, we have u ∈ H (Ω) and k u k H (Ω) ≤ C k f k L (Ω) . Sobolevembedding theorem says that u ∈ C ,α loc (Ω) and k u k C ,α (Ω ′ ) ≤ C k u k H (Ω ′ ) for anycompact subset Ω ′ . Thus, we get that ∆ u = u − f ∈ C ,α loc (Ω). It follows from Lemma4.2 and Theorem 4.6 in [16] that u ∈ C ,α loc (Ω) and, for any compact subsets Ω andΩ of Ω with Ω ⊂⊂ Ω , we have k u k C ,α (Ω ) ≤ C ( k f k C ,α (Ω ) + k u k C ,α (Ω ) ) ≤ C ( k f k C ,α (Ω ) + k u k H (Ω ) ) ≤ C ( k f k C ,α (Ω ) + k f k L (Ω ) ) , where C > and ∂ Ω . (cid:3) Local-in-time existence
This section proves that (1.1) has a unique local (in time) classical solution. Wemake use of the integral representation (2.5). Due to the difficulty of applying thecontraction mapping principle in the setting of H¨older spaces, the proof is dividedinto two steps. The first step applies the contraction mapping principle to (2.5) in thesetting of Sobolev spaces to obtain a unique local solution. The second step obtainsthe desired regularity of the local solution through a bootstrapping procedure.
Lemma 4.1.
Let ( g, h ) ∈ H ( R ) × C ([0 , T ]; H ( R )) , and φ satisfy the condition (1.5) . Then there is S with < S ≤ T , depending only on g and h , such that (2.5) has a unique solution v ∈ C ([0 , S ]; H ( R )) . D GBBM EQUATION 7
Proof.
This local existence and uniqueness result is proven through the contractionmapping principle. More precisely, we show that A defined in (2.5) is a contractionmap from B (0 , R ) ⊂ C ([0 , S ]; H ( R )) to itself, where B (0 , R ) denotes the closed ballcentered at 0 with radius R in C ([0 , S ]; H ( R )). S and R will be specified later inthe proof. It follows from (1.5), (2.5), Proposition 3.1 and the mean value theoremthat, for v, w ∈ B (0 , R ), kA v k C t H x ≤ k g k H + C k h k C t H x + CS k div( φ ( v + he − x )) k C t L x ≤ k g k H + C k h k C t H x + CS k φ ′ ( v + he − x ) k C t L ∞ x k∇ ( v + he − x ) k C t L x ≤ C + C S (1 + R ) R (4.1)and kA v − A w k C t H x ≤ CS k div( φ ( v + he − x )) − div( φ ( w + he − x )) k C t L x ≤ CS X j =1 Z R (cid:16)(cid:12)(cid:12) φ ′ j ( v + he − x )( v − w ) x j (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) φ ′ j ( v + he − x ) − φ ′ j ( w + he − x ) (cid:1) ( w + he − x ) x j (cid:12)(cid:12) (cid:17) dx ≤ CS n k φ ′ ( v + he − x ) k C t L ∞ x k v − w k C t H x + k φ ′′ ( v + he − x ) k C t L ∞ x k∇ ( w + he − x ) k C t L x k v − w k C t L x o ≤ C S (1 + R ) k v − w k C t H x , (4.2)where v lies between the line segment joining v and w , C is a constant depending on k g k H and k h k C t H x , and C is a constant depending on k h k C t H x . Note that (4.2)implies A is a continuous map of C ([0 , S ]; H ( R )) to itself. According to (4.1), A maps B (0 , R ) onto itself if R ≥ C + C S (1 + R ) R. (4.3)Hence, by (4.2), A is a contraction mapping of this ball if C S (1 + R ) <
1. Theseconditions will be met if we take R = 2 C and find a positive value S > C S (1 + 2 C ) ≤ . (4.4)Now, let v ( x, t ) = e g ( x ) + B h ( x, t )and v n ( x, t ) = A v n − ( x, t ) = v ( x, t ) + C v n − ( x, t ) for n ≥ . Y.-C. LIN, C. H. A. CHENG, J. M. HONG, J. WU, AND J.-M. YUAN
The contraction mapping principle gives that the sequence v n ( x, t ) converges in C ([0 , S ]; H ( R )) to the unique solution v of (2.5) in the ball k v k C t H x ≤ R . (cid:3) If the initial data g and the boundary data h are also H¨older, then the correspondingsolution can also be shown to be H¨older. This is achieved through the Sobolevembeddings and a bootstrapping procedure. Lemma 4.2.
Assume g ∈ C ,α loc ( R ) ∩ H ( R ) and h ∈ C ([0 , T ]; C ,α loc ( R ) ∩ H ( R )) for some < α < . Let φ satisfy the condition (1.5) . Then any solution v ∈ C ([0 , T ]; H ( R )) of (2.5) actually belongs to C ([0 , T ]; H ( R )) ∩ C ([0 , T ]; C ,α loc ( R )) .Proof. Since v ∈ C ([0 , T ]; H ( R )) and h ∈ C ([0 , T ]; H ( R )), we havediv( φ ( v + he − x )) ∈ C ([0 , T ]; H ( R )) . Proposition 3.1 gives(I − ∆) − { div( φ ( v + he − x )) } ∈ C ([0 , T ]; H ( R )) ֒ → C ([0 , T ]; C ,α ( R )) . (4.5)On the other hand, h ∈ C ([0 , T ]; C ,α loc ( R )) implies h x x t e − x ∈ C ([0 , T ]; C ,α loc ( R )) . Proposition 3.2 then yields(I − ∆) − { h x x t e − x } ∈ C ([0 , T ]; C ,α loc ( R )) . (4.6)In view of (4.5) and (4.6), we obtain v t = (I − ∆) − { h x x t e − x − div( φ ( v + he − x )) } ∈ C ([0 , T ]; C ,α loc ( R )) , which implies that v ( x, t ) = e g ( x ) + Z t v τ ( x, τ ) dτ = g ( x ) − h ( x , e − x + Z t v τ ( x, τ ) dτ ∈ C ([0 , T ]; C ,α loc ( R )) . (4.7)Using (4.7) and Proposition 3.2, we have v t = (I − ∆) − { h x x t e − x − div( φ ( v + he − x )) } ∈ C ([0 , T ]; C ,α loc ( R )) , and hence v ∈ C ([0 , T ]; C ,α loc ( R )). (cid:3) D GBBM EQUATION 9 Global-in-time existence
This section shows that the local (in time) solution obtained in the previous sectioncan be extended into a global one. This is achieved by establishing a global boundfor k v ( t ) k H under the condition that the flux φ obeys suitable growth condition. Westart with a global H -bound. Lemma 5.1.
Suppose ( g, h ) ∈ H ( R ) × C ([0 , S ]; H ( R )) , and φ satisfy the condi-tion (1.5) . Then the solution v of (2.5) obtained in Lemma satisfies the estimates k v k H ≤ k g k H + CS k h k C t H x (1 + k h k C t H x )+ C (1 + k h k C t H x ) Z t k v k H ds (5.1) and k v k H ≤ h k g k H + CS k h k C t H x (1 + k h k C t H x ) i / e CS (1+ k h k C t H x ) , (5.2) where C > is a constant depending only on φ .Proof. Multiplying (2.1a) by v and integrating over R , we get12 ddt Z R ( v + |∇ v | ) dx = Z R { h x x t e − x − div( φ ( v + he − x )) } v dx = Z R h x x t e − x v dx − Z R div( vφ ( v + he − x )) dx + Z R φ ( v + he − x ) · ∇ v dx. (5.3)Since v = 0 on ∂ R , Z R div( vφ ( v + he − x )) dx = Z ∂ R vφ ( v + he − x ) · n dS = 0 . (5.4)Now let Φ ∈ C ( R ; R ) satisfy Φ ′ = φ and Φ(0) = 0. Then Z R φ ( v + he − x ) · ∇ v dx = Z R φ ( v + he − x ) · ∇ ( v + he − x ) dx − Z R φ ( v + he − x ) · ∇ ( he − x ) dx = Z R div(Φ( v + he − x )) dx − Z R φ ( v + he − x ) · ∇ ( he − x ) dx = Z ∂ R Φ( h ) · n dS − Z R φ ( v + he − x ) · ∇ ( he − x ) dx. (5.5)By the Sobolev embedding H ( R ) ֒ → L q ( R ) for all q ≥ h ∈ C ([0 , S ]; H ( R )) ⊂ C ([0 , S ]; L q ( R )) . Employing the mean value theorem together with (1.5) and the properties of Φ, weobtain | Φ( h ) − Φ(0) | ≤ C ( | h | + | h | ) . Hence (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ R Φ( h ) · n dS (cid:12)(cid:12)(cid:12)(cid:12) = Z ∂ R | Φ( h ) − Φ(0) | dS ≤ C ( k h k C t L x + k h k C t L x ) ≤ C ( k h k C t H x + k h k C t H x ) . (5.6)Applying the mean value theorem and (1.5) again, we have | φ ( v + he − x ) | ≤ C ( | v + he − x | + | v + he − x | ) . As a consequence, (cid:12)(cid:12)(cid:12)(cid:12) Z R φ ( v + he − x ) · ∇ ( he − x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k v + he − x k L + k v + he − x k L ) k h k H x ≤ C ( k v k L + k h k L x + k v k L + k h k L x ) k h k H x ≤ C ( k v k H + k v k H + k h k C t H x + k h k C t H x ) k h k C t H x . (5.7)From (5.3)-(5.7), we can conclude that ddt k v k H ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z R h x x t e − x v dx (cid:12)(cid:12)(cid:12)(cid:12) + C ( k v k H + k h k C t H x ) k h k C t H x ≤ C ( k v k H + k v k H + k h k C t H x + k h k C t H x ) k h k C t H x ≤ C ( k v k H + k h k C t H x )(1 + k h k C t H x ) , that is, k v k H ≤ k g k H + CS k h k C t H x (1 + k h k C t H x ) + C (1 + k h k C t H x ) Z t k v k H ds, where C depends only on φ . Gronwall’s inequality gives k v k H ≤ {k g k H + CS k h k C t H x (1 + k h k C t H x ) } / e CS (1+ k h k C t H x ) . (cid:3) Now we derive the H -estimates based on the H -estimates we just obtained. Lemma 5.2.
Suppose ( g, h ) ∈ H ( R ) × h ∈ C ([0 , S ]; H ( R )) , and φ satisfy thecondition (1.5) . Then the solution v of (2.5) obtained in Lemma satisfies theestimate k v k H ≤ C (1 + S ) / exp n CS (1 + S ) / e CS o , where C > is a constant depending only on g , h and φ . D GBBM EQUATION 11
Proof.
Multiplying (2.1a) by ∆ u and then integrating on R , we have12 ddt Z R ( |∇ v | + | ∆ v | ) dx = Z R h x x t e − x ∆ v dx − Z R div( φ ( v + he − x ))∆ v dx. (5.8)By the mean value theorem and (1.5), | div( φ ( v + he x )) | ≤ | φ ′ ( v + he − x ) | |∇ ( v + he − x ) |≤ C (1 + | v + he − x | ) |∇ ( v + he − x ) | . (5.9)Thus, H¨older’s inequality gives ddt Z R ( |∇ v | + | ∆ v | ) dx ≤ k h x x t k L x k ∆ v k L + C k∇ ( v + he − x ) k L k ∆ v k L + C k v + he − x k L k∇ ( v + he − x ) k L k ∆ v k L . By the Sobolev embedding H ( R ) ֒ → L ( R ) and Young’s inequality, ddt ( k∇ v k L + k ∆ v k L ) ≤ C ( k h k C t H x + k h k C t H x )+ C (1 + k v k C t H x + k h k C t H x ) k v k H , where C > φ ; that is, k∇ v k L + k ∆ v k L ≤ C {k g k H + S ( k h k C t H x + k h k C t H x ) } + C (1 + k v k C t H x + k h k C t H x ) Z t k v k H ds. (5.10)Combining (5.1) and (5.10), we obtain k v k H ≤ C {k g k H + S k h k C t H x (1 + k h k C t H x ) } + C (1 + k v k C t H x + k h k C t H x ) Z t k v k H ds. (5.11)Applying (5.2) to (5.11), we have k v k H ≤ C (1 + S ) + C (1 + S ) / e CS Z t k v k H ds, where C depends only on g , h , and φ . Therefore, by Gronwall’s inequality, k v k H ≤ C (1 + S ) / exp n CS (1 + S ) / e CS o which concludes the proof of the lemma. (cid:3) Continuous dependence of the solution on data
This section is devoted to proving Theorem 1.2. That is, we establish the desiredcontinuous dependence. For the sake of clarity, we will divide the rest of this sectioninto two subsections. The first subsection proves the continuous dependence in theregularity setting of H while the second subsection focuses on the continuous depen-dence in the intersection space of H and a H¨older class. The precise statements canbe found in the lemmas below.6.1. Continuous dependence in H . Let L m denote the mapping that takes thedata g and h to the corresponding solutions of (1.1). By Theorem 1.1 we have L m : X m = H ( R ) × C ([0 , T ]; H ( R )) −→ C ([0 , T ]; H ( R )) . Since H ( R ) and C ([0 , T ]; H ( R )) are Banach spaces, the space X m equipped withthe usual product topology is also a Banach space. Lemma 6.1.
Suppose that φ ∈ C ( R , R ) satisfies the condition (1.5) . Then L m iscontinuous.Proof. Let ( g i , h i ) ∈ X m and u i = L m ( g i , h i ) be the mild solution of (1.1) correspond-ing to the initial data g i and the boundary data h i , i = 1 ,
2. Set v i = u i − h i e − x , i = 1 ,
2. Then v i satisfies the following initial-boundary value problem: ( v i ) t − ∆( v i ) t − ( h i ) x x t e − x + div ( φ ( v i + h i e − x )) = 0 ,v i ( x,
0) = g i ( x ) − h i ( x , e − x := e g i ( x ) ,v i (cid:0) ( x , , t (cid:1) = 0 . Define w = v − v . Then w satisfies: ( w t − ∆ w t − h x x t e − x + div ( φ ( v + h e − x )) − div ( φ ( v + h e − x )) = 0 ,w ( x,
0) = e g ( x ) , w (cid:0) ( x , , t (cid:1) = 0 , (6.1)where e g = e g − e g and h = h − h . In addition, we derive that w satisfies the followingintegral equation: w ( x, t ) = e g ( x ) + (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1) − Z t (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9) dτ. (6.2)Given ε >
0. Suppose that the distance between ( g , h ) and ( g , h ) in X m is smallenough such that(a) k e g k H ≤ ε , (b) k h k C t H x ≤ ε . D GBBM EQUATION 13
Taking H norm on both sides of (6.2) and using Proposition 3.1, we derive k w k H ≤ k e g k H + k h k C t H x + C Z t k div( φ ( v + h e − x )) − div( φ ( v + h e − x )) k L dτ. (6.3)Since div( φ ( v + h e − x )) − div( φ ( v + h e − x ))= φ ′ ( v + h e − x ) · ∇ ( v + h e − x ) − φ ′ ( v + h e − x ) · ∇ ( v + h e − x )= { φ ′ ( v + h e − x ) − φ ′ ( v + h e − x ) } · ∇ ( v + h e − x ) (6.4)+ φ ′ ( v + h e − x ) · ∇ ( w + he − x ) , the mean value theorem and condition (1.5) yield (cid:12)(cid:12) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:12)(cid:12) ≤ C {| w + he − x | · |∇ ( v + h e − x ) | + (1 + | v + h e − x | ) · |∇ ( w + he − x ) |} . (6.5)Applying (6.5) and H¨older’s inequality to (6.3), we obtain k w k H ≤ k e g k H + k h k C t H x + C Z t k w + he − x k L k∇ ( v + h e − x ) k L dτ + C Z t k∇ ( w + he − x ) k L dτ + C Z t k v + h e − x k L k∇ ( w + he − x ) k L dτ. By Sobolev’s inequality, k w k H ≤ k e g k H + k h k C t H x + C Z t k w + he − x k H dτ ≤ k e g k H + C (1 + T ) k h k C t H x + C Z t k w k H dτ, (6.6)where C depends only on v , v , h , h , and φ . Then Gronwall’s inequality gives k w k C t H x ≤ {k e g k H + k h k C t H x } e CT . (6.7)Note that w = v − v = u − u − ( h − h ) e − x = L m ( g , h ) − L m ( g , h ) − he − x . Therefore, by (6.7), kL m ( g , h ) − L m ( g , h ) k C t H x ≤ k w k C t H x + k h k C t H x ≤ {k e g k H + k h k C t H x } e CT ≤ e CT ε. (cid:3) Continuous dependence in the intersection of H and a H¨older space. This subsection proves the continuous dependence in the setting of the intersectionof H and a H¨oler space. First, we introduce the metrics on the spaces C ,α loc (Ω) and C ([0 , T ]; C ,α loc (Ω)), where Ω can be R or R . Let { Ω i } ∞ i =1 be an increasing sequenceof compact subsets of Ω satisfy(i) Ω i ⊂⊂ Ω i +1 for all i ∈ N ,(ii) ∞ [ i =1 Ω i = Ω.For a function f ∈ C ([0 , T ]; C ,α loc (Ω)), we define ρ i ( f ) = k f k C ([0 ,T ]; C ,α (Ω i )) for i ∈ N . Then { ρ i } forms a family of seminorms on C ,α loc (Ω). For f , f ∈ C ([0 , T ]; C ,α loc (Ω)),we define d ( f , f ) = ∞ X i =1 − i ρ i ( f − f )1 + ρ i ( f − f ) . Then d is a metric on C ([0 , T ]; C ,α loc (Ω)). It is clear that f k → f with respect to d if and only if ρ i ( f k − f ) → i . The topology induced by the metric d isindependent of the choice of the sequence { Ω i } ∞ i =1 . A metric on the space C ,α loc (Ω)can be defined similarly if we replace the seminorm above by ρ i ( f ) = k f k C ,α (Ω i ) for f ∈ C ,α loc (Ω) and i ∈ N .Now we fix a sequence { Ω i } ∞ i =1 of compact convex sets in R such that the con-ditions (i) and (ii) hold. Let I i denote the projection of Ω i to x -axis. Then { I i } ∞ i =1 forms a sequence of compact sets in R satisfying (i) and (ii). As stated above, the se-quences { Ω i } ∞ i =1 and { I i } ∞ i =1 induce metrics d on C ,α loc ( R ), d on C ([0 , T ]; C ,α loc ( R )),and d on C ([0 , T ]; C ,α loc ( R )) respectively. In Theorem 1.1, we get that for a givenpair ( g, h ) ∈ X c := [ H ( R ) ∩ C ,α loc ( R )] × C ([0 , T ]; H ( R ) ∩ C ,α loc ( R ))of initial and a boundary data, then (1.1) admits a unique classical solution u ∈ Y := C ([0 , T ]; H ( R )) ∩ C ([0 , T ]; C ,α loc ( R )) . If we let L c denote the mapping that takes the pair ( g, h ) into the correspondingclassical solution u , then L c : X c → Y. (6.8) D GBBM EQUATION 15
We remark that if (
M, d M ) and ( N, d N ) are two metric spaces, then the productspace M × N is a metric space with metric d M × N ( x, y ) = d M ( x, y ) + d N ( x, y ) for x, y ∈ M × N , and their intersection M ∩ N is a metric space with metric d M ∩ N ( x, y ) = d M ( x, y ) + d N ( x, y ) for x, y ∈ M ∩ N .We can immediately conclude that L c is a continuous mapping in (6.8) if we provethat i ◦ L c and j ◦ L c are both continuous where i and j are the natural inclusions of Y into C ([0 , T ]; H ( R )) and C ([0 , T ]; C ,α loc ( R )) respectively. Lemma 6.2.
Suppose that φ ∈ C ( R , R ) satisfies the conditions (1.5) - (1.6) . Then L c is continuous.Proof. By the discussions before the statement of this lemma, it suffices to prove L c : X c → C ([0 , T ]; H ( R )) and L c : X c → C ([0 , T ]; C ,α loc ( R )) (6.9)are both continuous. Comparing the metrics of the spaces X m and X c , we can easilyget the continuity of the mapping L c : X c → C ([0 , T ]; H ( R )) from Lemma 6.1.In this proof, we focus on showing that the second mapping of (6.9) is sequentiallycontinuous.Let ( g i , h i ) ∈ X c and u i = L c ( g i , h i ) be the classical solution of (1.1) correspondingto the initial data g i and the boundary data h i , i = 1 ,
2. Let w , e g , and h be definedas in the proof of Lemma 6.1. Then w satisfies the initial-boundary value problem(6.1), the integral equation (6.2), and the estimate (6.7). Let Ω, Ω ′ , and Ω ′′ be anygiven three compact convex sets in R with Ω ′′ ⊂⊂ Ω ′ ⊂⊂ Ω and let I and I ′ be theprojections of Ω and Ω ′ to the x -axis respectively. First, we take C ,α (Ω ′ ) norm onboth sides of (6.2) and use Sobolev’s inequality to obtain k w k C ,α (Ω ′ ) ≤ k e g k C ,α (Ω ′ ) + (cid:13)(cid:13) (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1)(cid:13)(cid:13) C ,α (Ω ′ ) + Z t (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) C ,α ( R ) dτ ≤ k e g k C ,α (Ω ′ ) + (cid:13)(cid:13) (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1)(cid:13)(cid:13) C ,α (Ω ′ ) (6.10)+ Z t (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) H ( R ) dτ. Lemma (3.2) implies (cid:13)(cid:13) (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1)(cid:13)(cid:13) C ,α (Ω ′ ) ≤ C n(cid:13)(cid:13) { h x x ( x , t ) − h x x ( x , } e − x (cid:13)(cid:13) C ,α (Ω) + (cid:13)(cid:13) { h x x ( x , t ) − h x x ( x , } e − x (cid:13)(cid:13) L (Ω) o ≤ C (cid:0) k h k C ,α ( I ) + k h k H ( I ) (cid:1) , (6.11)where C depends only on the distance between Ω ′ and ∂ Ω. Employing Proposition3.1 and applying mean value theorem together with (1.5) to (6.4), we derive that (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) H ( R ) ≤ C (cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) H ( R ) ≤ C n k w + he − x k W , ( R ) k∇ ( v + h e − x ) k L ( R ) + k w + he − x k L ∞ ( R ) k∇ ( v + h e − x ) k H ( R ) + k v + h e − x k W , ( R ) k∇ ( w + he − x ) k L ( R ) + (1 + k v + h e − x k L ∞ ( R ) ) k∇ ( w + he − x ) k H ( R ) o . Thus, Sobolev’s inequality gives (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) H ( R ) ≤ C (cid:16) k w k H ( R ) + k h k H ( R ) (cid:17) , (6.12)where C depends only on v , v , h , h , and φ . Combining the estimates (6.10)-(6.12),we get k w k C ,α (Ω ′ ) ≤ k e g k C ,α (Ω ′ ) + C (cid:0) k h k C ([0 ,T ]; C ,α ( I )) + k h k C ([0 ,T ]; H ( I )) (cid:1) + CT (cid:16) k w k C ([0 ,T ]; H ( R )) + k h k C ([0 ,T ]; H ( R )) (cid:17) ≤ k e g k C ,α (Ω ′ ) + C k h k C ([0 ,T ]; C ,α ( I )) + C (1 + T ) k h k C ([0 ,T ]; H ( R )) + CT k w k C ([0 ,T ]; H ( R )) . (6.13)Next, by taking C ,α (Ω ′′ ) norm on both sides of (6.2), we have k w k C ,α (Ω ′′ ) ≤ k e g k C ,α (Ω ′′ ) + (cid:13)(cid:13) (I − ∆) − (cid:0) { h x x ( x , t ) − h x x ( x , } e − x (cid:1)(cid:13)(cid:13) C ,α (Ω ′′ ) (6.14)+ Z t (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) C ,α (Ω ′′ ) dτ. D GBBM EQUATION 17
We use Proposition 3.2 to deduce that (cid:13)(cid:13) (I − ∆) − (cid:8) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:9)(cid:13)(cid:13) C ,α (Ω ′′ ) ≤ C (cid:16)(cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) C ,α (Ω ′ ) + (cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) L (Ω ′ ) (cid:17) . (6.15)For the estimate of the last term in the right hand side of (6.15), we use the proof ofLemma 6.1 to obtain (cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) L (Ω ′ ) ≤ C k w + he − x k H ( R ) . (6.16)In view of (6.4) and the convexity of Ω ′ , (cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) C ,α (Ω ′ ) ≤ (cid:13)(cid:13) φ ′ ( v + h e − x ) − φ ′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) (cid:13)(cid:13) ∇ ( v + h e − x ) (cid:13)(cid:13) C ,α (Ω ′ ) + (cid:13)(cid:13) φ ′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) (cid:13)(cid:13) ∇ ( w + he − x ) (cid:13)(cid:13) C ,α (Ω ′ ) . Since φ ∈ C ( R , R ) satisfies the conditions (1.5)-(1.6), (cid:13)(cid:13) φ ′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) ≤ C (1 + k v + h e − x k C (Ω ′ ) ) ≤ C and (cid:13)(cid:13) φ ′ ( v + h e − x ) − φ ′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) ≤ (cid:13)(cid:13) φ ′′ ( v + h e − x ) − φ ′′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) (cid:13)(cid:13) ∇ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) + (cid:13)(cid:13) φ ′′ ( v + h e − x ) (cid:13)(cid:13) C (Ω ′ ) (cid:13)(cid:13) ∇ ( w + he − x ) (cid:13)(cid:13) C (Ω ′ ) ≤ C (cid:13)(cid:13) w + he − x (cid:13)(cid:13) C ,α (Ω ′ ) , where C depends only on v , v , h , h , and φ . Thus, we have (cid:13)(cid:13) div( φ ( v + h e − x )) − div( φ ( v + h e − x )) (cid:13)(cid:13) C ,α (Ω ′ ) ≤ C (cid:13)(cid:13) w + he − x (cid:13)(cid:13) C ,α (Ω ′ ) ≤ C k e g k C ,α (Ω ′ ) + C k h k C ([0 ,T ]; C ,α ( I )) (6.17)+ C (1 + T ) k h k C ([0 ,T ]; H ( R )) + CT k w k C ([0 ,T ]; H ( R )) , where we used (6.13) in the last inequality. The estimates (6.11), (6.14)-(6.17) yield k w k C ([0 ,T ]; C ,α (Ω ′′ )) ≤ C (1 + T ) k e g k C ,α (Ω ′ ) + C (1 + T ) k h k C ([0 ,T ]; C ,α ( I )) + C (1 + T ) k h k C ([0 ,T ]; H ( R )) + CT (1 + T ) k w k C ([0 ,T ]; H ( R )) . It follows from (6.7) that k w k C ([0 ,T ]; C ,α (Ω ′′ )) ≤ e CT n k e g k C ,α (Ω ′ ) + k e g k H ( R ) + k h k C ([0 ,T ]; C ,α ( I )) + k h k C ([0 ,T ]; H ( R )) o . (6.18) Finally, let { ( g k , h k ) } ∞ k =1 be a sequence of X c that converges to ( g , h ) in X c .Suppose that u k = L c ( g k , h k ), k ∈ N ∪ { } be the corresponding classical solutions of(1.1) with respect to the initial data g k and the boundary data h k . Set, for k ∈ N ∪{ } , ( v k = u k − h k e − x , e g k = g k − h k e − x . We define, for k ∈ N , w k = v k − v , e g k, = e g k − e g , h k, = h k − h . Since { ( g k , h k ) } ∞ k =1 converges to ( g , h ) in X c , we have(a) d ( e g k , e g ) → k e g k − e g k H ( R ) → d ( h k , h ) → k h k − h k C ([0 ,T ]; H ( R )) → ′ ) k e g k, k C ,α (Ω i ) → i ∈ N ,(b ′ ) k e g k, k H ( R ) → ′ ) k h k, k C ([0 ,T ]; C ,α ( I i )) → i ∈ N ,(d ′ ) k h k, k C ([0 ,T ]; H ( R )) → i ∈ N , applying the estimate (6.18), we get k w k k C ([0 ,T ]; C ,α (Ω i )) ≤ e CT n k e g k, k C ,α (Ω i +1 ) + k e g k, k H ( R ) + k h k, k C ([0 ,T ]; C ,α ( I i +1 )) + k h k, k C ([0 ,T ]; H ( R )) o → , and hence kL c ( g k , h k ) − L c ( g , h ) k C ([0 ,T ]; C ,α (Ω i )) ≤ k w k k C ([0 ,T ]; C ,α (Ω i )) + k h k, k C ([0 ,T ]; C ,α (Ω i )) ≤ e CT n k e g k, k C ,α (Ω i +1 ) + k e g k, k H ( R ) + k h k, k C ([0 ,T ]; C ,α ( I i +1 )) + k h k, k C ([0 ,T ]; H ( R )) o → , which implies that d ( L c ( g k , h k ) , L c ( g , h )) → L c : X c → C ([0 , T ]; C ,α loc ( R )) is sequentiallycontinuous. (cid:3) D GBBM EQUATION 19 Results for the GBBM-Burgers equation
The purpose of this section is to generalize the above results to the 2D GBBM-Burgers equation, that is, equation (1.1) with ν = 1. The results established inprevious sections also hold for the GBBM-Burgers equation.The proofs of Theorems 1.1 and 1.2 for the case when ν = 1 are essentially thesame as those for the case when ν = 0. In fact, as in the case when ν = 0, werewrite equation (1.1) as(I − ∆) v t + ∆ v + div (cid:0) φ ( v + he − x ) (cid:1) = e he − x in R × (0 , T ) , (7.1a) v = e g on R × { t = 0 } , (7.1b) v = 0 on ∂ R × (0 , T ) , (7.1c)where e g is again given by (2.2) and e h is defined by e h ( x, t ) = h x x t ( x, t ) − h x x ( x, t ) − h ( x, t ) . (7.1a) can be modified as v + v t = (I − ∆) − { v + e he − x − div( φ ( v + he − x )) } (7.2)which suggest that v ( x, t ) = e − t e g ( x ) + Z t e − ( t − s ) (I − ∆) − { v + e he − x − div( φ ( v + he − x )) } ds = e − t e g ( x ) + Z t e − ( t − s ) (I − ∆) − (cid:8) ( h x x s − h x x − h ) e − x (cid:9) ds (7.3)+ Z t e − ( t − s ) (I − ∆) − { v − div( φ ( v + he − x )) } ds . Based on the fact that e − ( t − s ) is bounded by 1 for s ∈ [0 , t ], exactly the same procedureof proving the existence of a unique solution (using the contraction mapping principle)for the case ν = 0 can be applied to yield the results corresponding to Theorems 1.1and 1.2. Acknowledgments.
AC was supported by the National Science Council (NSC)of Taiwan under grant 100-2115-M-008-009-MY3, MH was supported by NSC undergrant 101-2115-M-008-005, YL was supported by NSC under grant 102-2115-M-008-001, JW was supported by NSF under grant DMS 1209153, and JY was supportedby NSC under grant 101-2115-M-126-002.
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Ying-Chieh Lin, C. H. Arthur Cheng, and John M. HongDepartment of MathematicsNational Central UniversityChung-Li, Taiwan 32001, ROC
E-mail address : [email protected]; [email protected]; [email protected] Jiahong Wu
D GBBM EQUATION 21
Department of MathematicsOklahoma State University401 Mathematical SciencesSillwater, OK 74078, USA
E-mail address : [email protected] Juan-Ming YuanDepartment of Financial and Computational MathematicsProvidence UniversityTaichung, Taiwan 43301, ROC
E-mail address ::