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Heat flow of biharmonic maps in dimensions four and its application
- PURE APPL. MATH. Q
, 2007
"... Let (M, g) be a four dimensional compact Riemannian manifold without boundary, (N, h) ⊂ Rk be a compact Riemannian submanifold without boundary. We establish the existence of a global weak solution to the heat flow of extrinsic biharmonic maps from M to N, which is smooth away from finitely many ..."
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Cited by 13 (7 self)
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Let (M, g) be a four dimensional compact Riemannian manifold without boundary, (N, h) ⊂ Rk be a compact Riemannian submanifold without boundary. We establish the existence of a global weak solution to the heat flow of extrinsic biharmonic maps from M to N, which is smooth away from finitely many singular times. As a consequence, we prove that if Π4(N) = {0}, then any free homotopy class α ∈ [M, N] contains at least one minimizing biharmonic map.
Energy identity of approximate biharmonic maps to Riemannian . . .
, 2011
"... We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in Lp for p> 4 3. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivia ..."
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Cited by 4 (0 self)
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We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in Lp for p> 4 3. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on R4. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity.
Well-posedness for the heat flow of biharmonic maps with rough initial data
, 2010
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Regularity and uniqueness of the heat flow of biharmonic maps
"... In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere S L ⊂ R L+1 under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian ..."
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Cited by 1 (1 self)
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In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere S L ⊂ R L+1 under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian energy, and unique limit at t = ∞. We establish both regularity and uniqueness for Serrin’s (p, q)-solutions to the heat flow of biharmonic maps into any compact Riemannian manifold N without boundary. 1
US
, 2009
"... into spheres in the critical dimension, Annales de l’Institut Henri Poincaré- Analyse non linéaire ..."
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into spheres in the critical dimension, Annales de l’Institut Henri Poincaré- Analyse non linéaire
EQUIVARIANT POLYHARMONIC MAPS
"... Abstract. We study O(d)-equivariant polyharmonic maps and their associ-ated heat flows. We are mainly interested in blowup phenomena for the higher order flows. Such results have been hard to prove, due to the inapplicability of the maximum principle to these higher order flows. We believe that the ..."
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Abstract. We study O(d)-equivariant polyharmonic maps and their associ-ated heat flows. We are mainly interested in blowup phenomena for the higher order flows. Such results have been hard to prove, due to the inapplicability of the maximum principle to these higher order flows. We believe that the ideas developed herein could be useful in the study of other higher order parabolic equations. We prove that blowup occurs in the biharmonic map heat flow from B(0, 1; 4) into S4. To our knowledge, this was the first example of blowup in the higher order polyharmonic map heat flows. We provide Mathematica code that computes our symmetry reduction for the polyharmonic map heat flow of any order. This code is then used to explicitly compute our symmetry reduc-tions for the harmonic, as a check, and biharmonic cases. Next, the possible O(d)-equivariant biharmonic maps from R4 into S4 are classified. Finally, we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from B(0, 1; 4) into S4 that wind around S4 as many times as we wish. 1.
