Non-uniqueness of weak solutions to the wave map problem

Widmayer, Klaus

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 519-532

In this note we show that weak solutions to the wave map problem in the energy-supercritical dimension 3 are not unique. On the one hand, we find weak solutions using the penalization method introduced by Shatah [12] and show that they satisfy a local energy inequality. On the other hand we build on a special harmonic map to construct a weak solution to the wave map problem, which violates this energy inequality.Finally we establish a local weak-strong uniqueness argument in the spirit of Struwe [15] which we employ to show that one may even have a failure of uniqueness for a Cauchy problem with a stationary solution. We thus obtain a result analogous to the one of Coron [2] for the case of the heat flow of harmonic maps.

Classification : 35L05, 35L71
Keywords: Wave maps, Weak solutions, Weak-strong uniqueness

[1] Piotr Bizoń, Tadeusz Chmaj, Zbisław Tabor, Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere, Nonlinearity 14 no. 5 (2001), 1041 -1053 | MR | Zbl

[2] J.-M. Coron, Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 no. 4 (1990), 335 -344 | MR | EuDML | Zbl | Numdam

[3] Piero D'Ancona, Vladimir Georgiev, Wave maps and ill-posedness of their Cauchy problem, New Trends in the Theory of Hyperbolic Equations, Oper. Theory Adv. Appl. vol. 159 , Birkhäuser, Basel (2005), 1 -111 | MR | Zbl

[4] Roland Donninger, On stable self-similar blowup for equivariant wave maps, Commun. Pure Appl. Math. 64 no. 8 (2011), 1095 -1147 | MR | Zbl

[5] A. Freire, Global weak solutions of the wave map system to compact homogeneous spaces, Manuscr. Math. 91 no. 4 (1996), 525 -533 | MR | EuDML | Zbl

[6] Pierre Germain, Besov spaces and self-similar solutions for the wave-map equation, Commun. Partial Differ. Equ. 33 no. 7–9 (2008), 1571 -1596 | MR | Zbl

[7] Pierre Germain, On the existence of smooth self-similar blowup profiles for the wave map equation, Commun. Pure Appl. Math. 62 no. 5 (2009), 706 -728 | MR | Zbl

[8] Frédéric Hélein, Harmonic Maps, Conservation Laws and Moving Frames, Camb. Tracts Math. vol. 150 , Cambridge University Press, Cambridge (2002) | MR | Zbl

[9] S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 no. 1 (1996), 99 -133 | MR | Zbl

[10] Sergiu Klainerman, Sigmund Selberg, Remark on the optimal regularity for equations of wave maps type, Commun. Partial Differ. Equ. 22 no. 5–6 (1997), 901 -918 | MR | Zbl

[11] Nader Masmoudi, Fabrice Planchon, Unconditional well-posedness for wave maps, J. Hyperbolic Differ. Equ. 9 no. 2 (2012), 223 -237 | MR | Zbl

[12] Jalal Shatah, Weak solutions and development of singularities of the $\mathrm{SU}\left(2\right)$ σ-model, Commun. Pure Appl. Math. 41 no. 4 (1988), 459 -469 | MR | Zbl

[13] Jalal Shatah, Michael Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. 7 no. 303ff (1994) | MR | Zbl

[14] Jalal Shatah, Michael Struwe, Geometric Wave Equations, Courant Lect. Notes Math. vol. 2 , New York University Courant Institute of Mathematical Sciences, New York (1998) | MR | Zbl

[15] Michael Struwe, Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality, Commun. Pure Appl. Math. 52 no. 9 (1999), 1179 -1188 | MR | Zbl

[16] Terence Tao, Global regularity of wave maps. II. Small energy in two dimensions, Commun. Math. Phys. 224 no. 2 (2001), 443 -544 | MR | Zbl

[17] Daniel Tataru, Rough solutions for the wave maps equation, Am. J. Math. 127 no. 2 (2005), 293 -377 | MR | Zbl