Abstract
We discuss the \((1+1)\)-dimensional wave maps equation with values in a compact Riemannian manifold . Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting combines analytic, geometric, and probabilistic methods.
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Notes
To be precise, [AD99] constructed the Wiener measure on the compact domain [0, 1] instead of \({\mathbb {R}}\).
In dimension \(d=1\), the renormalization in (1.7) is not necessary.
For technical reasons, we will later write the linear evolution as \(\phi _{\text {lin} }=\theta (\phi ^+(u)+\phi ^{-}(v))\), where \(\theta >0\) is a small parameter. Due to this, we will need to adjust the definitions of \(\phi ^+\) and \(\phi ^-\), see e.g. (3.2). In the introduction, we ignore this technicality.
When \(N>M\), the roles of \(\phi ^+\) and \(\phi ^-\) should be reversed.
While the dual space of \(L^\infty _x\) is not \(L^1_x\), one can still characterize the \(L^\infty _x\)-norm as a supremum over integrals against \(L^1_x\)-normalized functions. This statement generalizes to our functions spaces and is sufficient for the duality argument used here.
While it is customary to write \(B_x\) or \(W_x\) for Brownian motions indexed by the variable x, we use the notation B(x), respectively W(x), which we consider to be more in line with the notation in the rest of the paper.
Since B and \(\bar{W}\) are independent and hence have vanishing cross-variation, the Itô and Stratonovich integrals are identical in this case.
This is the shortest argument which yields an acceptable contribution. By using frequency-support considerations for \(A_M^-\) and \(A_N^+\), we can obtain better decay in M and N, and thus this term is less serious than our argument suggests.
While \(H_M^+\) is not necessarily \(A_M^+\), the same argument applies.
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Acknowledgements
The authors thank Rishabh Gvalani, Florian Kunich, Stephan Luckhaus, Felix Otto, Igor Rodnianski, Angela Stevens, Terence Tao, and Markus Tempelmayr for helpful and interesting discussions. The authors also thank the anonymous referees for valuable comments and suggestions. B.B. thanks the MPI for Mathematics in the Sciences for support during a visit in the summer of 2021. The three authors thank ICERM, which is supported by NSF grant DMS-1929284, for support during the semester program on Hamiltonian Methods in Dispersive and Wave Evolution Equations. B.B. was partially supported by the NSF under Grant No. DMS-1926686. J.L. was partially supported by NSF grant DMS-1954707. G.S. was partially supported by DMS-1764403, DMS-2052651 and the Simons Foundation.
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Appendix A. Deterministic Ill-Posedness
Appendix A. Deterministic Ill-Posedness
We now prove the mild ill-posedness statement in Theorem 1.2, which concerns the unboundedness of the first Picard iterate.
Proof of Theorem 1.2.(ii)
We only treat the case \({\mathbb {S}}^2\), since the argument easily generalizes to \({\mathbb {S}}^{D-1}\) with \(D\geqslant 3\). We split the argument into two steps. In the first step, we present several reductions which simplify the first Picard iterate. In the second step, we construct an explicit sequence of functions for which the first Picard iterate diverges.
Step 1: Reductions. We let \(\phi _0,\phi _1\in C^\infty _b({\mathbb {R}}\rightarrow {\mathbb {R}}^3)\) satisfy
Here, \(\Vert \cdot \Vert _2\) refers to the Euclidean norm on \({\mathbb {R}}^3\). Then, the right and left-moving linear waves are given by
We recall that the second fundamental form of the sphere \({\mathbb {S}}^2\) is given by
As a result, the first Picard iterate of the wave maps equation (WM) is given by
We now fix any time \(t>0\). Furthermore, we let \(\chi \) be our previous nonnegative, smooth cut-off function, which satisfies \({\text {supp}}(\chi ) \subseteq (-3,3)\), and let \(0<\epsilon \leqslant t/100\). For any \(r\in {\mathbb {R}}\), it then holds that
In order to prove the unboundedness of the first Picard iterate on \(C^r\times C^{r-1}\) for any \(r\leqslant 1/2\), it therefore suffices to prove that
We now choose \(\phi _0(x)=e_3 \in {\mathbb {R}}^3\) and choose \(\phi _1(x)=\psi ^\prime (x)\) for \(\psi (x) \in C^\infty _c((-1,1)\rightarrow {\mathbb {R}}^3)\). In particular, it holds that \(\Vert \psi \Vert _{C^{1/2}} \lesssim \Vert \phi _1 \Vert _{C^{-1/2}}\). Due to the geometric constraint in (A.1), \(\psi \) has to satisfy \(\langle e_3, \psi \rangle =0\). Using our assumptions, the first Picard iterate takes the form
In order to further simplify , we assume that \({\text {supp}}(\psi ) \subseteq (-2\epsilon ,2\epsilon )\). In particular, it holds that \(\psi (x+t)=\psi (x-t)=0\) for all \(x\in (-2\epsilon ,2\epsilon )\). Then, a direct computation yields that
As a result, we obtain that
We now write \(\psi (y)=\psi ^1(y) e_1 +\psi ^2(y) e_2\), which satisfies the constraint \(\langle \psi ,e_3 \rangle =0\). Then, the first coordinate of is given by
Thus, it suffices to prove that
Step 2: Proof of (A.3). The main idea in the proof of (A.3) is to create a severe high\(\times \)high\(\times \)low-interaction in the integrand. In order to cover the endpoint \(C^{1/2}\), however, we need to be careful and work at multiple scales.
We let \(\kappa ,\kappa _0 \in {\mathbb {N}}\) be arbitrary and define the set of frequencies
Then, we define
Since the frequencies in \(F_{\kappa ,\kappa _0}\) are well-separated, it holds that
where the implicit constant is uniform in \(\kappa \) and \(\kappa _0\). We now claim that
Before proving (A.6), we first show that (A.6) implies (A.3). By integrating (A.6) against \(\chi (x/\epsilon )\), using that \(1-\cos (2y)\geqslant 0\), and using that \(0<\epsilon \leqslant t/10\), we obtain
where \(C_\epsilon >0\) and \(c_\epsilon >0\) are sufficiently large and small constants, respectively. We now obtain the desired conclusion (A.3) by first choosing a parameter \(\kappa _0=\kappa _0(\epsilon )\) such that \(C_\epsilon 2^{-5\kappa _0}\) is smaller than \( c_\epsilon \) and then letting \(\kappa \rightarrow \infty \). Thus, it now only remains to prove the claim (A.6). By inserting (A.4) and (A.5) into the integrand, we obtain that
where we have already estimated terms in which the derivative hits the cut-off \(\chi (y/\epsilon )\). We start by analyzing the main term (A.7). First, we treat the contribution of the diagonal case \(m=n\). Using trigonometric identities, we have that
Using integration by parts, this yields
which is the main term in (A.6). In the non-diagonal case \(m\ne n\), we use that the frequencies in \(F_{\kappa ,\kappa _0}\) are well-separated. Together with integration by parts, this yields
We now estimate the first error term (A.8). Using integration by parts, it follows that
It remains to estimate the third error term (A.9). To this end, we distinguish two cases. In the case \(\max (\ell ,m,n)>{{\,\textrm{med}\,}}(\ell ,m,n)\), we use that the frequencies in \(F_{\kappa ,\kappa _0}\) are well-separated, which implies that
In the case \(\max (\ell ,m,n)={{\,\textrm{med}\,}}(\ell ,m,n)\), we only use that \(\sin (\cdot )\) and \(\cos (\cdot )\) are bounded, which implies that
This completes the proof of (A.6). \(\square \)
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Bringmann, B., Lührmann, J. & Staffilani, G. The Wave Maps Equation and Brownian Paths. Commun. Math. Phys. 405, 60 (2024). https://doi.org/10.1007/s00220-023-04885-5
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DOI: https://doi.org/10.1007/s00220-023-04885-5