aa r X i v : . [ m a t h . A P ] F e b ENERGY BOUNDS FOR BIHARMONIC WAVE MAPS IN LOWDIMENSIONS
TOBIAS SCHMID
Abstract.
For compact, isometrically embedded Riemannian manifolds
N ֒ → R L , weintroduce a fourth-order version of the wave map equation. By energy estimates, we provean priori estimate for smooth local solutions in the energy subcritical dimension n = 1 , n = 1 ,
2. We also give a proof of the uniqueness of solutions thatare bounded in these Sobolev norms. Introduction
Let (
N, h ) be a (compact) Riemannian manifold, isometrically embedded (by Nash’stheorem) into euclidean space
N ֒ → R L . For a Riemannian manifold ( M, g ), we introducethe action functional L ( u ) = 12 Z T Z M | ∂ t u ( x, t ) | − | ∆ g ( x ) u ( x, t ) | dV g ( x ) dt, dV g = p det g dx for (smooth) maps u : M × [0 , T ) → N . We call u a (extrinsic) biharmonic wave map, if L is critical in the following sense. ddδ L ( u + δ Φ) | δ =0 = 0 , Φ ∈ C ∞ c ( M × [0 , T ) , R L ) , Φ( x, t ) ∈ T u ( x,t ) N, ( x, t ) ∈ M × [0 , T ) . In this case, u satisfies the condition(1.1) ∂ t u ( x, t ) + ∆ g u ( x, t ) ⊥ T u ( x,t ) N, ( x, t ) ∈ M × [0 , T ) , where ∆ g denotes the Laplace-Beltrami operator on ( M, g ). More explicitly, we use the factthat there exists a smooth familiy of orthogonal (linear) projector P p : R L → T p N, p ∈ N, in order to expand (1.1) into the equation ∂ t u + ∆ g u = ( I − P u )( ∂ t u + ∆ g u ) = dP u ( ∂ t u, ∂ t u ) + ∆ g (tr g dP u ( ∇ u, ∇ u ))(1.2) + 2 div g ( dP u ( ∇ u, ∆ g u )) − dP u (∆ g u, ∆ g u ) . The projector maps are derivatives of the metric distance (with respect to N ) in R L , ie. p = Π( p ) + 12 ∇ p (dist ( p, N )) , P p = ∇ p Π( p ) , dist( p, N ) < δ . We note that via this representation, it is possible to extend this family smoothly to allof R L in order to solve the Cauchy problem for (1.2) without restricting the coefficients apriori. Mathematics Subject Classification.
Primary: 35L75. Secondary: 58J45.
Key words and phrases. biharmonic, fourth-order wave equation, energy estimates, global solutions.The author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) – Project-ID 258734477 – SFB 1173.
We are particularly interested in the following Cauchy problem for M = R n ∂ t u ( t, x ) + ∆ u ( t, x ) ⊥ T u ( t,x ) N, ( t, x ) ∈ (0 , T ) × R n ( u (0 , x ) , u t (0 , x )) = ( u ( x ) , u ( x )) , x ∈ R n ( u , u ) : R n → T N, u ( x ) ∈ T u ( x ) N x ∈ R n (1.3)We state the following result Theorem 1.1.
Let n ∈ { , } and u ∈ C ∞ ( R n × [0 , T ) , N ) be a local solution of (1.3) .Assume further u − u ∈ C ([0 , T ) , H n +2 ( R n )) ∩ C ([0 , T ) , H n ( R n )) . Then there holds (1.4) lim sup t ր T ( k u t ( t ) k H n + k∇ u ( t ) k H n +1 ) < ∞ , as long as T < ∞ . In the recent work [6], the authors prove local wellposedness (in high regularity) and ablow up condition for the Cauchy problem (1.3), which (by the proof of Theorem 1.1) impliesthe following
Corollary 1.2.
Let n ∈ { , } and u , u : R n → R L , u ( x ) ∈ N, u ( x ) ∈ T u ( x ) N for x ∈ R n and such that ( ∇ u , u ) ∈ H k ( R n ) × H k − ( R n ) , for k ∈ N with k ≥ n + 1 . Then the Cauchy problem (1.3) has a global solution u : R n × R → N with u − u ∈ C ( R , H k +1 ( R n )) ∩ C ( R , H k − ( R n )) . In particular, if u , u are smooth and supp ( ∇ u ) , supp ( u ) are compact, then there exists aglobal smooth solution of (1.3) . This work is part of the authors PhD thesis [7]. We conclude this section with a fewremarks.In the sense explained above, (1.1) and (1.2) are higher order versions of the wave mapequation(1.5) (cid:3) g u = tr g dP u ( ∇ u, ∇ u ) , with the d’Alembert operator (cid:3) g = ∂ t − ∆ g . Equation (1.5) is the Euler Lagrange equationof the action functional L ( u ) = Z T Z M L ( u ) dV g dt on the Riemannian manifold ( M, g ) with Lagrangian L ( u ) = ˜ g αβ h ∂u∂x α , ∂u∂x β i , and where ∂∂x = ∂∂t and ˜ g = − dt + g . This wave equation has been studied intensively in the past,especially as a model problem for nonlinear dispersion and singularity formation. We referto [9] and [3] for an overview over the wellposedness and singularity theory of the Cauchyproblem for the wave map equation (1.5).For the wave map problem, the action functional L is independent of the embedding N ֒ → R L . In our case however, there is an intrinsic biharmonic wave map problem, arisingfrom critical points of the (embedding independent) functional L i ( u ) = Z T Z M | ∂ t u | h − | tr g ( ∇ du ) | h dV g dt. IHARMONIC WAVE MAPS 3 where ∇ denotes the Levi-Civita connection of the pullback bundle u ∗ T N endowed with thepullback metric u ∗ h and the energy potential is given by the tension field τ g ( u ) = tr g ( ∇ du )of u . Moreover, first variations are calculated intrinsically as follows. ddδ L i ( u δ ) | δ =0 = 0 , u · ∈ C ∞ (( − δ , δ ) × M × [0 , T ) , N ) , u = u such that supp( u − u δ ) ⊂⊂ M × (0 , T ) for | δ | < δ .Then the Euler-Lagrange equation, which has been calculated for static solutions e.g. in [4],becomes(1.6) ∇ t ∂ t u + ∆ g,h u + R ( u )( du, ∆ g,h u ) du = 0 , where R is the curvature tensor and in the covariant notation, we set ∆ g,h u = tr g ( ∇ du ),and use ∆ g,h u = ∆ g,h (∆ g,h u ) = tr g ( ∇ τ g ( u )).Static solutions of (1.2) (and (1.6)) are extrinsic (and intrinsic) biharmonic maps, i.e. theyare maps u : ( M, g ) → ( N, h ) between Riemannian manifolds that are critical for the(intrinsic or extrinsic) energy functional F ( u ) = 12 Z M | tr g ( ∇ du ) | h dV g , E ( u ) = 12 Z M | ∆ g u | h dV g , respectivelywhere the latter is defined subject to an isometric embedding ( N, h ) ֒ → R m . Biharmonicmaps (resp. the Euler Lagrange equation of E and F ) and their heat flow has been studiedintensively in the past.2. Related work and local wellposedness in high regularity
In [5], the authors pove the existence of a global weak solution into round spheres S L − ⊂ R L .This is done by a penalization functional of Ginzburg Landau type, which then gives a uni-form energy bound in the penalty parameter. To prove convergence of such approximations,the authors depend on the geometry of the sphere, more precisely, the equation can berewritten in divergence form. This argument has been used for the wave map equation (1.5)with N = S L − and M = R n in [8] and further the divergence form has been used in [10], inorder to prove weak compactness of the class of stationary solutions of (1.2) on the domain M = R .As mentioned above, in the recent work [6], the authors prove local wellposedness of theCauchy problem (1.3). More precisely, let u , u : R n → R L , u ( x ) ∈ N, u ( x ) ∈ T u ( x ) N for L n a.e. x ∈ R n with( ∇ u , u ) ∈ H k − ( R n ) × H k − ( R n ) , k > j n k + 2 , k ∈ N . Then there exists a
T > u : R n × [0 , T ) → N of (1.3) with u − u ∈ C ([0 , T ) , H k ( R n )) ∩ C ([0 , T ) , H k − ( R n )) . From this, we note that in particular we obtain Corollary 1.2 from a blow up conditioncontained in [6]. In the following, note that the energy functional(2.1) E ( u ( t )) = 12 Z R n | u t | + | ∆ u | dx, is formally conserved along solutions u . This implies the bound(2.2) ddt Z R n |∇ u | dx ≤ E ( u (0)) . T.SCHMID
Both, (2.1) and (2.2), will be used in the following for smooth solutions. We further notethat below in section 4, we inculde a short argument for the uniqueness of such solutions indimension n = 1 , ,
3. 3.
Proof of Theorem 1.1
Since for solutions u of (1.2), resp. the Cauchy problem (1.3), the term ∂ t u + ∆ u isa section over the normal bundle of u ∗ ( T N ), we let codim( N ) = L − l for l ∈ N , l ≤ L and first assume the normalbundle T ⊥ N of N ⊂ R L is parallelizable. This means thereexists a frame of (smooth) orthogonal vectorfields { ν ( p ) , . . . , ν L − l ( p ) } ⊂ R L , p ∈ N with ν i ( p ) ⊥ T p N for every p ∈ N .In this case, for any local solution u , we have an explicit representation for the nonlinearityin terms of ν i ( u ).(3.1) ∂ t u + ∆ u =: L − l X i =1 G i ( u ) ν i ( u ) =: G i ( u ) ν i ( u ) , where G i ( u ) = h ∂ t u + ∆ u, ν i ( u ) i . We thus calculate h ∂ t u, ν i ( u ) i = −h u t , dν i ( u ) u t i , h ∆ u, ν i ( u ) i = − h∇ ∆ u, dν i ( u ) ∇ u i − h∇ u, dν i ( u ) ∇ ∆ u i− h∇ u, d ν i ( u )( ∇ u ) + 2 d ν i ( u )( ∇ u, ∇ u ) + d ν i ( u )( ∇ u, ∆ u ) i− h∇ u, d ν i ( u )( ∇ u ) + dν i ( ∇ u ) i − h ∆ u, d ν i ( u )( ∇ u ) + dν i (∆ u ) i , where we denote by d k ν i the kth order differential of ν i on N and write ( ∇ u ) , ( ∇ u ) forproducts of first order derivatives of u with eiter two order three factors, respectively. Theprecise product, e.g. ∂ x j u · ∂ x j u or ∂ x i u · ∂ x j u · ∂ x j u will become clear in the terms of theexpansion. The result in Theorem 1.1 is known for N = S L − and n ≤ Case: n = 2
We apply ∆ = ∂ i ∂ i on both sides of (3.1). Then, testing the differentiatedequation by ∆ u t , we infer d dt Z R n ( | ∆ u t | + | ∆ u | ) dx = Z R n ∆( G i ( u ) ν i ( u ))∆ u t dx. (3.2)Since G i ( u ) contains derivatives of order three, we can not proceed by the H¨older inequality.Instead, we follow [2], where the authors showed that the highest order derivative cancel inthe case N = S L − , ν ( u ) = u . Since∆( G i ( u ) ν i ( u ))∆ u t = ∆( G i ( u )) ν i ( u )∆ u t + 2 ∇ ( G i ( u )) · ∇ ( ν i ( u ))∆ u t + G i ( u )∆ ν i ( u )∆ u t , and 0 = ∆( ν i ( u ) u t ) = 2 dν i ( u )( ∇ u ) · ∇ u t + ν i ( u )∆ u t + d ν i ( u )( ∇ u ) u t + dν i ( u )(∆ u ) u t , it follows∆( G i ( u ) ν i ( u ))∆ u t = − ∆ G i ( u ) (cid:0) dν i ( u )( ∇ u ) · ∇ u t + d ν i ( u )( ∇ u ) u t + dν i ( u )(∆ u ) u t (cid:1) + 2 ∇ G i ( u ) · dν i ( u )( ∇ u )∆ u t + G i ( u ) (cid:0) d ν i ( u )( ∇ u ) + dν i ( u )∆ u (cid:1) ∆ u t . IHARMONIC WAVE MAPS 5
Hence we observe, by integration by parts for the first summand, Z R n ∆( G i ( u ) ν i ( u ))∆ u t dx = Z R n ∇ G i ( u ) · [3 d ν i ( u )( ∇ u ) ∇ u t + 3 dν i ( u )(∆ u ) ∇ u t ] dx + Z R n ∇ G i ( u ) · [4 dν i ( u )( ∇ u )∆ u t + d ν i ( u )( ∇ u ) u t ] dx + Z R n ∇ G i ( u ) · [3 d ν i ( u )(∆ u, ∇ u ) u t + dν i ( u )( ∇ ∆ u ) u t ] dx + Z R n G i ( u )( d ν i ( u )( ∇ u ) u t + dν i ( u )(∆ u ))∆ u t dx. Instead of deducing bounds for this terms that depend on the normal frame { ν , . . . ν L − l } ,we turn to the general case and use the normal projector I − P u : R L → ( T u N ) ⊥ along themap u : R n × [0 , T ) → N in order to represent the nonlinearity in (3.1) as(3.3) ∂ t u + ∆ u = ( I − P u )( ∂ t u + ∆ u ) . Here, we proceed similarly, ie. we use∆(( I − P u )( ∂ t u + ∆ u ))∆ u t = ∆(( I − P u ) ( ∂ t u + ∆ u ))∆ u t , (3.4)and hence∆(( I − P u ) ( ∂ t u + ∆ u ))∆ u t =∆[( I − P u )](( I − P u )( ∂ t u + ∆ u ))∆ u t + 2 ∇ ( I − P u ) · ∇ (( I − P u )( ∂ t + ∆ u ))∆ u t + (∆[( I − P u )( ∂ t u + ∆ u )])( I − P u )∆ u t . In order to treat the last summand, we expand0 = ∆(( I − P u ) u t ) = ( I − P u )∆ u t − d P u (( ∇ u ) , u t ) − dP u (∆ u, u t ) − dP u ( ∇ u, ∇ u t ) . Hence, as before, integration by parts yields Z R n ∆(( I − P u )( ∂ t u + ∆ u ))∆ u t = − Z R n d P u (( ∇ u ) , ( I − P u )( ∂ t u + ∆ u ))∆ u t dx − Z R n dP u (∆ u, ( I − P u )( ∂ t u + ∆ u ))∆ u t dx − Z R n dP u ( ∇ u, ∇ [( I − P u )( ∂ t u + ∆ u )])∆ u t dx − Z R n ∇ [( I − P u )( ∂ t u + ∆ u )] · ∇ [ dP u (∆ u, u t ) + 2 dP u ( ∇ u, ∇ u t ) + d P u (( ∇ u ) , u t )] dx. We first note the pointwise bounds | ( I − P u )( ∂ t u + ∆ u ) | . | u t | + |∇ u | + |∇ u ||∇ u | + |∇ u ||∇ u | + |∇ u | (3.5) |∇ [( I − P u )( ∂ t u + ∆ u )] | . |∇ u t || u t | + |∇ u || u t | + | ∆ u ||∇ u | (3.6) + |∇ u | ( |∇ u | + |∇ u | ) + |∇ u ||∇ u | + |∇ u | |∇ u | + |∇ u | , where the constants only depend on the supremum norm k dP k C b = k dP k C b ( N ) + (cid:13)(cid:13) d P (cid:13)(cid:13) C b ( N ) + (cid:13)(cid:13) d P (cid:13)(cid:13) C b ( N ) + (cid:13)(cid:13) d P (cid:13)(cid:13) C b ( N ) . T.SCHMID
We now estimate, using (3.5) and (3.6), k d P u (( ∇ u ) , ( I − P u )( ∂ t u + ∆ u ))∆ u t k L . k ∆ u t k L k∇ u k L ∞ (cid:20) k u t k L ∞ k u t k L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ k ∆ u k L + k∇ u k L ∞ k ∆ u k L + k∇ ∆ u k L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:21) , k dP u (∆ u, ( I − P u )( ∂ t u + ∆ u ))∆ u t k L . k ∆ u t k L k ∆ u k L ∞ (cid:20) k u t k L ∞ k u t k L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ k ∆ u k L + k∇ u k L ∞ k ∆ u k L + k∇ ∆ u k L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:21) = k ∆ u t k L k ∆ u k L ∞ (cid:2) k u t k L ∞ k u t k L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ k ∆ u k L (cid:3) + h ( t ) k ∆ u t k L k ∆ u k L ∞ (cid:2) k ∆ u k L + k∇ u k L (cid:3) + h ( t ) k ∆ u t k L k ∆ u k L ∞ k∇ ∆ u k L , where we set h ( t ) := k∇ u ( t ) k L ∞ . We further note that the equality is up to the constantfrom the estimate and hence proceed by estimating k dP u ( ∇ u, ∇ [( I − P u )( ∂ t u + ∆ u )])∆ u t k L . k ∆ u t k L k∇ u k L ∞ (cid:20) k u t k L ∞ k∇ u t k L + k∇ u k L k u t k L ∞ + (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L k∇ u k L ∞ + k∇ ∆ u k L ( (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ u k L ∞ ) + k∇ u k L ∞ k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k ∆ u k L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:21) . The latter upper bound equals the sum of h ( t ) k ∆ u t k L (cid:2) k u t k L ∞ k∇ u t k L + k∇ u k L k u t k L ∞ + k∇ ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k ∆ u k L k∇ u k L ∞ (cid:3) , and h ( t ) k ∆ u t k L (cid:2) (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L + k∇ u k L ∞ k∇ ∆ u k L + k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ u k L k∇ u k L ∞ (cid:3) . We calculate ∇ [ dP u (∆ u, u t ) + 2 dP u ( ∇ u, ∇ u t ) + d P u (( ∇ u ) , u t )]= d P u ( ∇ u, ∆ u, u t ) + dP u ( ∇ ∆ u, u t ) + dP u (∆ u, ∇ u t ) + 2 d P u (( ∇ u ) , ∇ u t )+ 2 dP u ( ∇ u, ∇ u t ) + 2 dP u ( ∇ u, ∇ u t ) + d P u (( ∇ u ) , u t )+ 2 d P u ( ∇ u, ∇ u, u t ) + d P u (( ∇ u ) , ∇ u t ) , and hence k∇ [ dP u (∆ u, u t ) + 2 dP u ( ∇ u, ∇ u t ) + d P u (( ∇ u ) , u t )] · ∇ [( I − P u )( ∂ t u + ∆ u )] k L . (cid:0) k ∆ u k L k∇ u k L ∞ k u t k L ∞ + k∇ ∆ u k L k u t k L ∞ + ( k ∆ u k L ∞ + k∇ u k L ∞ ) k∇ u t k L + k ∆ u t k L k∇ u k L ∞ + k∇ u k L ∞ k u t k L (cid:1)(cid:2) k u t k L ∞ k∇ u t k L + k∇ u k L k u t k L ∞ + (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L k∇ u k L ∞ + k∇ ∆ u k L ( (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ u k L ∞ ) + k∇ u k L ∞ k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ . + k ∆ u k L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:3) . IHARMONIC WAVE MAPS 7
We now collect all terms which are quadratic, linear or constant in h ( t ), i.e. the latter boundequals J ( u ) + h ( t ) J ( u ) + h ( t ) J ( u ) + h ( t ) J ( u ) , where J ( u ) = ( k∇ ∆ u k L k u t k L ∞ + k ∆ u k L ∞ k∇ u t k L ) (cid:2) k u t k L ∞ k∇ u t k L + k∇ u k L k u t k L ∞ + k∇ ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k ∆ u k L k∇ u k L ∞ (cid:3) ,J ( u ) = ( k∇ ∆ u k L k u t k L ∞ + k ∆ u k L ∞ k∇ u t k L ) (cid:2) (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L + k∇ u k L ∞ k∇ ∆ u k L + k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ u k L k∇ u k L ∞ (cid:3) ,J ( u ) = ( k ∆ u k L k u t k L ∞ + k∇ u t k L k∇ u k L ∞ + k ∆ u t k L + k∇ u k L ∞ k u t k L ) (cid:2) k u t k L ∞ k∇ u t k L , + k∇ u k L k u t k L ∞ + k∇ ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k ∆ u k L k∇ u k L ∞ (cid:3) ,J ( u ) = ( k ∆ u k L k u t k L ∞ + k∇ u t k L k∇ u k L ∞ + k ∆ u t k L + k∇ u k L ∞ k u t k L ) (cid:2) (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L + k∇ u k L ∞ k∇ ∆ u k L + k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ u k L k∇ u k L ∞ (cid:3) . We note that the energy is conserved, ie. for t ∈ [0 , T )(3.7) 2 E ( u ( t )) = k ∆ u ( t ) k L + k ∂ t u ( t ) k L = k ∆ u k L + k u k L = 2 E ( u , u ) , and further, this implies the boundssup t ∈ [0 ,T ) k∇ u ( t ) k L . √ T ( p E ( u , u ) + k∇ u k L ) , and(3.8) sup t ∈ [0 ,T ) k u ( t ) − u k L . T p E ( u , u ) . (3.9)We recall the following cases of Gagliardo-Nirenberg’s interpolation for n = 2 k ∆ u k L ∞ + k∇ ∆ u k L . (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L k ∆ u k L , k u t k L ∞ . k ∆ u t k L k u t k L , (3.10) k∇ u k L ∞ . (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L k∇ u k L , k∇ u k L . (cid:13)(cid:13) ∆ u (cid:13)(cid:13) L k∇ u k L , and(3.11) k∇ u t k L . k ∆ u t k L k u t k L . (3.12)Setting E ( u ( t )) := k ∆ u t ( t ) k L + (cid:13)(cid:13) ∆ u ( t ) (cid:13)(cid:13) L , t ∈ [0 , T ) , by (3.10), (3.11) and the estimates above, there exists a constant C ( T ) = C ( N, u , u )(1 + T ) α for some α >
0, such that C ( N, u , u ) only depends on the norm k dP k C b , the optimalSobolev constant in Gagliardo-Nirenberg’s interpolation and E ( u , u ) , k∇ u k L and suchthat the following holds. ddt E ( u ( t )) ≤ C ( T )(1 + h ( t ) + h ( t ))( E ( t ) + E ( t ))(3.13) ≤ C ( T )(1 + h ( t ))(1 + E ( t )) , t ∈ [0 , T ) . Using the idea from [2], we now apply the sharp Sobolev inequality of Brezis-Gallouet-Wainger from [1] in order to bound (we assume u is not a constant)(3.14) h ( t ) ≤ ˜ C k∇ u ( t ) k H k∇ u ( t ) k H k∇ u ( t ) k H !! , t ∈ [0 , T ) . Thus, using (3.10), (3.8) and (3.7),(3.15) h ( t ) ≤ C ( T ) (cid:0) (cid:0) E ( t ) (cid:1)(cid:1) , t ∈ [0 , T ) , T.SCHMID and hence ddt ( e + E ( u ( t ))) ≤ C ( T ) log (cid:0) e + E ( t ) (cid:1) ( e + E ( t )) , t ∈ [0 , T ) . (3.16)This suffices for a Gronwall-type inequality for log( e + E ( t )) and hence by (3.7) and (3.10),(3.11) and (3.12), we have lim sup t → T ( k u t k H + k∇ u k H ) < ∞ , as long as T < ∞ . Case: n = 1
Here, by Gagliardo-Nirenberg’s estimate, we infer the bound k∇ u k L ∞ . (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k∇ u k L . (3.17)Hence, the a priori bound is derived similarly for ( ∇ u ( t ) , u t ( t )) ∈ H ( R ) × H ( R ). We note ddt Z R |∇ u t | + |∇ ∆ u | dx = − Z R dP u ( ∇ u, ( I − P u )( ∂ t u + ∆ u )) · ∇ u t − Z R ( I − P u )( ∂ t u + ∆ u )) · ( d P u (( ∇ u ) , u t ) + dP u ( ∇ u, u t ) + dP u ( ∇ u, ∇ u t )) dx. Thus we estimate, as before k dP u ( ∇ u, ( I − P u )( ∂ t u + ∆ u )) ∇ u t k L . k∇ u t k L k∇ u k L ∞ (cid:20) k u t k L ∞ k u t k L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L + k∇ u k L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:21) , and k ( d P u (( ∇ u ) , u t ) + dP u ( ∇ u, u t ) + dP u ( ∇ u, ∇ u t ))[( I − P u )( ∂ t u + ∆ u )] k L . ( k u t k L k∇ u k L ∞ + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k u t k L ∞ + k∇ u t k L k∇ u k L ∞ ) (cid:20) k u t k L ∞ k u t k L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L + k∇ u k L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ (cid:21) Hence from the interpolation estimates (3.17), (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ . (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L , k u t k L ∞ . k∇ u t k L k u t k L , (3.18)and (3.7), (3.8), there holds (for C ( T ) > ddt (1 + E ( t )) ≤ C ( T )(1 + E ( t )) , t ∈ [0 , T )which suffices to use a Gronwall argument in order to conclude the proof.4. A uniqueness argument
We now give a short argument for the uniqueness of solutions u : R n × [0 , T ) → N, n =1 , , u − u (0) ∈ C ([0 , T ) , H ( R n )) ∩ C ([0 , T ) , H ( R n ))Setting w = u − v for solutions u, v of (1.3) in the class (4.1) with u (0) = v (0) , u t (0) = v t (0),we provide a Gronwall type argument in the energy space, i.e. more precisely for the norm k w t k L + k w k H . We note the interpolation estimate k w k L ∞ . k ∆ w k n L k w k − n L , n = 1 , , , IHARMONIC WAVE MAPS 9 and the identity d dt (cid:18)Z R n | w t | + | ∆ w | (cid:19) dx = I + I + I , (4.2)where I = Z R n (cid:2) dP u ( u t u t + 4 ∇ u · ∇ ∆ u + ∆ u ∆ u + 2 ∇ u · ∇ u ) + d P u ( ∇ u ) + d P u (2( ∇ u ) ∆ u + 4( ∇ u ) · ∇ u ) (cid:3) ( P v − P u ) w t dx + Z R n (cid:2) ( dP u − dP v )( u t u t + 4 ∇ u · ∇ ∆ u + ∆ u ∆ u + 2 ∇ u · ∇ u )+ ( d P u − d P v )( ∇ u ) + ( d P u − d P v )(2( ∇ u ) ∆ u + 4( ∇ u ) · ∇ u ) (cid:3) ( P u − P v ) u t dxI = Z R n (cid:2) dP v ( u t w t + w t v t + 4 ∇ w · ∇ ∆ u + ∆ w ∆ u + ∆ v ∆ w + 2 ∇ w · ∇ u + 2 ∇ v · ∇ w ) + d P v ( ∇ w · ( ∇ u ) + ∇ w · ( ∇ u ) ∇ v + ∇ w · ( ∇ v ) ∇ u + ∇ w · ( ∇ v ) )+ d P v (2 ∇ w · ∇ u ∆ u + 2 ∇ v · ∇ w ∆ u + 2( ∇ v ) ∆ w + 4 ∇ w · ∇ u · ∇ u + 4 ∇ v · ∇ w · ∇ u + 4( ∇ v ) · ∇ w (cid:3) ( P u − P v ) u t dxI = Z R n dP v ( ∇ v, ∇ ∆ w )( P u − P v ) u t dx. This follows from( I − P u )( ∂ t u − ∆ u ) − ( I − P v )( ∂ t v − ∆ v )= ( P v − P u )[( I − P u )( ∂ t u − ∆ u )]+ ( I − P v )[( I − P u )( ∂ t u − ∆ u ) − ( I − P v )( ∂ t v − ∆ v )] , and ( I − P v ) w t = ( I − P v ) u t = ( P u − P v ) u t . We further note Z R n dP v ( ∇ v, ∇ ∆ w )( P u − P v ) u t dx = − Z R n (cid:2) d P v (( ∇ v ) , ∆ w ) + dP v (∆ v, ∆ w ) (cid:3) ( P u − P v ) u t dx − Z R n dP v ( ∇ v, ∆ w )( P u − P v ) ∇ u t + dP v ( ∇ v, ∆ w )( dP u − dP v )( ∇ u, u t ) dx − Z R n dP v ( ∇ v, ∆ w ) dP v ( ∇ w, u t ) dx. Hence, we estimate I . ( k w t k L k w k L ∞ + k w k L ∞ k u t k L )( k u t k L k u t k L ∞ + k∇ ∆ u k L k∇ u k L ∞ + k∇ u k L k∇ u k L ∞ + k ∆ u k L k∇ u k L ∞ + k ∆ u k L (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ ) ,I . k w k L ∞ k u t k L ( k∇ w k L k∇ u k L ∞ + k∇ w k L k∇ v k L ∞ k∇ u k L ∞ + k∇ w k L k∇ v k L ∞ k∇ u k L ∞ + k∇ w k L k∇ v k L ∞ + k∇ w k L k∇ u k L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ w k L k∇ v k L ∞ (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ + k∇ v k L ∞ k ∆ w k L + max {k u t k L ∞ , k v t k L ∞ } k w t k L + max { (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ , (cid:13)(cid:13) ∇ v (cid:13)(cid:13) L ∞ } k ∆ w k L )+ k w k L ∞ k u t k L ∞ k∇ ∆ u k L k∇ w k L and I . k w k L ∞ k∇ u t k L k ∆ w k L k∇ v k L ∞ + k w k L ∞ k u t k L k ∆ w k L k∇ v k L ∞ k∇ u k L ∞ + k ∆ w k L k∇ w k L k u t k L ∞ k∇ v k L ∞ + k w k L ∞ k u t k L k ∆ w k L k∇ v k L ∞ + k w k L ∞ k u t k L ∞ k ∆ w k L k ∆ v k L . We set E ( t ) := k w t k L + k w k H . Using the aforementioned interpolation inequality, we obtain in particular k w k L ∞ . E ( t ).Since also ddt Z R n |∇ w | dx ≤ k w t k L + k ∆ w k L ≤ E ( t ) , and(4.3) ddt Z R n | w | dx ≤ k w k L + k w t k L ≤ E ( t ) , (4.4)estimating (4.2) gives ddt E ( t ) . (1 + k∇ u k H + k u t k H + k∇ v k H + k v t k H ) E ( t ) =: C ( u, v ) E ( t )(4.5)This suffices for uniqueness, as long as C ( u, v ) stays bounded in time. We also remark thatin n = 1, in order to conclude uniqueness from similar arguments, it suffices for a smoothsolution u to stay bounded in u ( t ) ∈ H ( R ) , ∂ t u ( t ) ∈ H ( R ). References [1] Br´ezis, H. and Wainger, S.
A note on limiting cases of Sobolev embeddings and convolution inequalities ,Communications in Partial Differential Equations, Taylor & Francis, Vol 5., No. 7, pp. 773-789 (1980).[2] Fan, J. and Ozawa, T.
On regularity criterion for the 2D wave maps and the 4D biharmonic wave maps ,In: Current advances in nonlinear analysis and related topics, GAKUTO Internat. Ser. Math. Sci. Appl.,Vol 32., pp. 69-83, Gakkotosho, Tokyo (2010).[3] Geba, D.-A. and Grillakis, M. G.
An introduction to the theory of wave maps and related geometricproblems , World Scientific (2017).[4] Guoying, J. , Note di Matematica, Vol.28, pp. 209-232 (2008).[5] Herr, S. and Lamm, T. and Schnaubelt, R.
Biharmonic wave maps into spheres , Proceedings of theAmerican Mathematical Society, Vol. 148, No. 2, pp. 787-796 (2020).[6] Herr, S. and Lamm, T. and Schmid, T. and Schnaubelt, R.
Biharmonic wave maps: local wellposednessin high regularity , Nonlinearity, IOP Publishing, Vol. 33, No. 5 (2020).[7] Schmid, T.
Local wellposedness and global regularity results for biharmonic wave maps , doi:https://publikationen.bibliothek.kit.edu/1000128147 (2021).[8] Shatah, J.
Weak solutions and development of singularities of the SU (2) σ -model , Communications onpure and applied mathematics, Wiley Online Library, Vol. 41, No. 4, 459-469 (1988).[9] Shatah, J. and Struwe, M. Geometric wave equations , American Mathematical Society, Providence, RI(1998), MR 1674843.[10] Strzelecki, P.
On biharmonic maps and their generalizations , Calculus of Variations and Partial Differ-ential Equations, Vol. 18, No. 4, pp. 401-432 (2003).
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