Global well-posedness for the Schr\"odinger map problem with small Besov norm

TL;DR

This paper proves that the Schr"odinger map problem has a globally well-posed solution when the initial data has a small Besov norm at the critical regularity level.

Contribution

It establishes global well-posedness for the Schr"odinger map problem under small initial data in a critical Besov space, advancing understanding of the problem's behavior.

Findings

Global well-posedness established for small Besov norm initial data.

Results apply at critical regularity levels.

Advances the mathematical theory of Schr"odinger maps.

Abstract

In this paper we prove a global result for the Schr\"odinger map problem with initial data with small Besov norm at critical regularity.

Equations

$${\partial }_t\varphi$$

$$\varphi (0)$$

$$\varphi ({\lambda }^{2}t,\lambda x)$$

$$H_Q^{\sigma }=\{f:{\mathbf{R}}^d\to {\mathbf{R}}^3,|f(x)|\equiv 1a.e.f-Q\in H^{\sigma }\},$$

$$d_Q^{\sigma }(f,g)=\Vert f-g{\Vert }_{H^{\sigma }({\mathbf{R}}^d)}.$$

$$H^{\infty }={\cap }_{\sigma \in {\mathbf{Z}}_{+}}{H}^{\sigma },\text{and}{H}_Q^{\infty }={\cap }_{\sigma \in {\mathbf{Z}}_{+}}{H}_Q^{\sigma }.$$

$$\Vert f{\Vert }_{{\dot{H}}^{\sigma }}=\Vert \mathcal{F}(f)(\xi )\cdot |\xi {|}^{\sigma }{\Vert }_{L^2},$$

$$\Vert f{\Vert }_{{\dot{H}}_Q^{\sigma }}=\Vert f-Q{\Vert }_{{\dot{H}}^{\sigma }},$$

$$\mathcal{F}(f)(\xi )=c_d{\int }_{{\mathbf{R}}^d}{e}^{-ix\cdot \xi }f(x)dx.$$

$$\varphi =S_Q({\varphi }_0)\in C(\mathbf{R}:H_Q^{\infty })$$

$$\underset{t\in \mathbf{R}}{\sup }\Vert \varphi (t)-Q{\Vert }_{{\dot{H}}^{d/2}}\le C\Vert {\varphi }_0-Q{\Vert }_{{\dot{H}}^{d/2}},$$

$$\underset{t\in [-T,T]}{\sup }\Vert \varphi (t){\Vert }_{H_Q^{\sigma }}\le C(\sigma ,T,\Vert \varphi {\Vert }_{H_Q^{\sigma }}).$$

$$\underset{t\in \mathbf{R}}{\sup }\Vert \varphi (t)-Q{\Vert }_{H^{\sigma }}\le C_{\sigma }\Vert {\varphi }_0-Q{\Vert }_{H^{\sigma }},\frac{d}{2}\le \sigma \le {\sigma }_1.$$

$$B_{{ϵ}_0}^{\sigma }=\{\varphi \in {\dot{H}}_Q^{d/2-1}\cap {\dot{H}}^{\sigma }:\Vert \varphi -Q{\Vert }_{{\dot{H}}^{d/2}}\le {ϵ}_0\}$$

$$E_0(t)={\int }_{{\mathbf{R}}^d}|\varphi (t)-Q{|}^{2}dx,$$

$$E_1(t)={\int }_{{\mathbf{R}}^d}\sum _{m=1}^d|{\partial }_m\varphi (t,x){|}^{2}dx.$$

$$\underset{j}{\sup }\Vert \psi (2^j\xi )|\xi {|}^{d/2}\cdot \mathcal{F}(\varphi -Q)(\xi ){\Vert }_{L^2({\mathbf{R}}^d)}\le {ϵ}_0(d,\Vert {\varphi }_0{\Vert }_{{\dot{H}}^{d/2}}).$$

$$i{u}_t+\Delta u=0.$$

$$\Vert |\nabla {|}^{\frac{3-d}{2}}|u{|}^2{\Vert }_{L_{t,x}^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\lesssim \Vert u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert u{\Vert }_{L_t^{\infty }{\dot{H}}_x^{1/2}(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$$\int {\int }_{x_1=y_1}|{\partial }_1(u(t,x_1,...,x_d)\overline{v(t,x_1,y_2,...,y_d)}){|}^{2}dxdydt$$

$$\lesssim \Vert u{\Vert }_{L_t^{\infty }{\dot{H}}^{1/2}(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert v{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2+\Vert v{\Vert }_{L_t^{\infty }{\dot{H}}^{1/2}(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$${\psi }_j(x)=\psi (2^{-j}x)-\psi (2^{-j+1}x),$$

$$\mathcal{F}(P_{j}f)(\xi )=[\psi (2^{-j}\xi )-\psi (2^{-j+1}\xi )]\mathcal{F}(f)(\xi ).$$

$$(P_{j}f)(x)=2^{jd}\int \check{\psi }(2^j(x-y))f(y)dy,$$

$$|\check{\psi }(x)|{\lesssim }_N(1+|x|{)}^{-N}.$$

$$\int {\int }_{x_1=y_1}|{\partial }_1(P_{j}u(t,x_1,x_2,...,x_d)\overline{P_{k}v(t,x_1,y_2,...,y_d)}){|}^{2}dxdydt$$

$$\lesssim \Vert P_{j}u{\Vert }_{L_t^{\infty }{\dot{H}}^{1/2}(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert P_{k}v{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2+\Vert P_{k}v{\Vert }_{L_t^{\infty }{\dot{H}}^{1/2}(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert P_{j}u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$$\Vert {\partial }_1(P_{j}u\overline{P_{k}v}){\Vert }_{L_{t,x}^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\lesssim 2^{(d-1)k}{2}^j\Vert P_{j}u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert P_{k}v{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$$\Vert \nabla (P_{j}u\overline{P_{k}v}){\Vert }_{L_{t,x}^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\lesssim 2^{(d-1)k}{2}^j\Vert P_{j}u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert P_{k}v{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$$\Vert (P_{j}u\overline{P_{k}v}){\Vert }_{L_{t,x}^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\lesssim 2^{(d-1)k}{2}^{-j}\Vert P_{j}u{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2\Vert P_{k}v{\Vert }_{L_t^{\infty }{L}_x^2(\mathbf{R}\times {\mathbf{R}}^d)}^2.$$

$$M(t)=\int |v(t,y){|}^2\frac{(x-y{)}_1}{|(x-y{)}_1|}Im[\bar{u}{\partial }_{1}u](t,x)dxdy$$

Full text

Global well-posedness for the Schrödinger map problem with small Besov norm

Benjamin Dodson

Abstract: In this paper we prove a global result for the Schrödinger map problem with initial data with small Besov norm at critical regularity.

1 Introduction

In this paper we consider the Schrödinger map initial value problem

[TABLE]

in dimensions $d\geq 3$, where $\phi:\mathbf{R}^{d}\times\mathbf{R}\rightarrow S^{2}\hookrightarrow\mathbf{R}^{3}$ is a continuous function. The Schrödinger map equation arises in ferromagnetism as the Heisenberg model for the ferromagnetic spin system.

In this paper we are concerned with the issue of global well - posedness of $(\ref{1.1})$ for data which is small in a critical space under the scaling.

Observe that for any $\lambda>0$, if $\phi(t,x)$ solves $(\ref{1.1})$, then

[TABLE]

also solves $(\ref{1.1})$. Now for $Q\in S^{2}$, $\mathbf{Z}_{+}=\{0,1,2,...\}$, and $\sigma\in[0,\infty)$, define the space

[TABLE]

where $H^{\sigma}$ is the usual inhomogeneous Sobolev space. This metric has the induced distance

[TABLE]

Also define the metric spaces

[TABLE]

The scaling law $(\ref{1.2})$ preserves the $\dot{H}^{d/2}$ and $\dot{H}_{Q}^{d/2}$ homogeneous Sobolev norms, where

[TABLE]

[TABLE]

and $\mathcal{F}:L^{2}\rightarrow L^{2}$ is the usual Fourier transform

[TABLE]

$(\ref{1.1})$ is globally well - posed for data which is small in $\dot{H}^{d/2}$.

Theorem 1.1** **(Global regularity)

Assume $d\geq 2$ and $Q\in S^{2}$. Then there exists $\epsilon_{0}(d)>0$ such that for any $\phi_{0}\in H_{Q}^{\infty}$ with $\|\phi_{0}-Q\|_{\dot{H}^{d/2}}\leq\epsilon_{0}(d)$, then there is a unique solution

[TABLE]

of the initial value problem $(\ref{1.1})$. Moreover,

[TABLE]

and for any $T\in[0,\infty)$ and $\sigma\in\mathbf{Z}_{+}$,

[TABLE]

This theorem was proved in dimensions $d\geq 4$ in [2], and then for dimensions $d\geq 2$ in [3]. [3] also proved a uniform global bound.

Theorem 1.2** **(Uniform bounds and well - posedness)

Assume $d\geq 2$, $Q\in S^{2}$, and $\sigma_{1}\geq\frac{d}{2}$. Then there exists $0<\epsilon_{0}(d,\sigma_{1})\leq\epsilon_{0}(d)$ such that if $\phi\in H_{Q}^{\infty}$ with $\|\phi-Q\|_{\dot{H}^{d/2}}\leq\epsilon_{0}(d,\sigma_{1})$, then the global solution $\phi(t)$ to $(\ref{1.1})$ with initial data $\phi_{0}$ satisfies

[TABLE]

In addition, the solution operator admits a continuous extension from

[TABLE]

to $C(\mathbf{R};\dot{H}^{\sigma}\cap\dot{H}_{Q}^{d/2-1})$.

The case when $d=2$ is particularly interesting, due to the fact that a solution to $(\ref{1.1})$ conserves the quantities

[TABLE]

and

[TABLE]

However, in general the proofs in theorems 1.1 and 1.2 are more difficult in lower dimensions. Indeed, [3] utilized the caloric gauge of [18] to analyze dimension $d=2$, because the Coulomb gauge used in [3] was not strong enough.

In this paper we will extend theorems 1.1 and 1.2 to data which is small in Besov - type norms.

Theorem 1.3

Suppose $\phi_{0}\in\dot{H}^{d/2}$ and suppose that $\psi(x)\in C_{0}^{\infty}(\mathbf{R}^{d})$ is a radial function supported on $\frac{1}{4}\leq|x|\leq 4$ and $\phi(x)=1$ on $\frac{1}{2}\leq|x|\leq 2$. Furthermore, suppose that for some $\epsilon_{0}(d,\|\phi_{0}\|_{\dot{H}^{d/2}})$,

[TABLE]

Then the results of theorems 1.1 and 1.2 hold.

In this paper we follow [1], [2], [3], [10], [11], [12], and [14] and prove a priori bounds on the derivative of the Schrödinger map rather than the Schrödinger map itself. We use the Coulomb gauge, as is typically used in analysis of higher dimensions, see for example [2].

The main new ingredient to this paper is the bilinear estimates obtained from the interaction Morawetz estimates. In [7], applying the bilinear estimates of [13] to the various Schrödinger map gauges gave bilinear estimates for solutions to $(\ref{1.1})$ in dimensions $d\geq 2$. These results were used in [8] to improve the results of [15], [16], and [17].

Here we are able to use bilinear arguments in order to prove theorem 1.3. These arguments greatly simplify the function spaces that are needed in the analysis.

2 Bilinear Virial estimates

The work in this paper will heavily rely on bilinear estimates for solutions to the linear Schrödinger equation

[TABLE]

Theorem 2.1** **(Interaction Morawetz estimate)

If $u$ solves $(\ref{2.1})$ then

[TABLE]

Proof: This theorem was proved in [6] when $d=3$. [19] subsequently extended the result to dimensions $d\geq 4$. [13] and [5] independently proved theorem 2.1 in dimensions $d=1,2$. $\Box$

Of course, [6], [19], [13], and [5] also proved theorem 2.1 for nonlinear Schrödinger equations as well. [13] also made some very important insights into the behavior of linear solutions as well. In particular, [13] observed that one may compute the interaction Morawetz estimate for two different solutions to $(\ref{2.1})$, yielding a bilinear estimate.

Theorem 2.2** **(Bilinear virial estimate)

Suppose $u$ and $v$ are solutions to $(\ref{2.1})$. Then

[TABLE]

In particular, suppose that $u$ and $v$ are solutions to $(\ref{2.1})$ and $P_{j}$ is the usual Littlewood - Paley projection operator. That is, let $\psi(x)=1$ for $|x|\leq 1$ and $\psi(x)=0$ for $|x|>2$ be a smooth, radially symmetric function, decreasing as $r\rightarrow\infty$. Then let

[TABLE]

and let $P_{j}$ be the Fourier multiplier given by

[TABLE]

Then by elementary Fourier analysis,

[TABLE]

where $\check{\psi}$ is a smooth function that is rapidly decreasing for $|x|$ large, that is, for any integer $N$,

[TABLE]

Since the Littlewood - Paley operator is a Fourier multiplier, if $u$ solves $(\ref{2.1})$ then $P_{j}u$ also solves $(\ref{2.1})$, so

[TABLE]

But then by $(\ref{2.6})$, $(\ref{2.7})$, $(\ref{2.8})$, and Hölder’s inequality, if $k\leq j$,

[TABLE]

Now since there is absolutely nothing special about the direction $e_{1}\in S^{d-1}$, averaging over all directions $\omega\in S^{1}$ gives

[TABLE]

Thus, if $j\geq k+10$, then by the Fourier support of $P_{j}u\cdot\overline{P_{k}v}$, and Bernstein’s inequality,

[TABLE]

Also, by theorem 1.1, when $d\geq 3$, and $j-10\leq k\leq j$,

[TABLE]

Remark: This will not work in dimensions $d=1,2$ because we can merely say that $P_{j}u\cdot\overline{P_{k}v}$ is supported on $|\xi|\lesssim 2^{k}$ when $j-10\leq k\leq j$, so when the power of $|\nabla|^{\frac{3-d}{2}}$ is positive, then $(\ref{2.12})$ does not follow directly from theorem 1.1 and the above analysis. It is for this technical reason that this paper only addresses dimensions $d\geq 3$.

It is appropriate to point out that $(\ref{2.11})$ has been proven to be true in [4] using Fourier analysis. In fact, [4] and subsequent Fourier analytic proofs in higher dimensions all predate the bilinear virial arguments of [13]. However, in this case, it is quite useful to examine the bilinear estimates through the lens of the virial identity. In [13] the interaction Morawetz for theorem 2.2 is

[TABLE]

Averaging over all directions in $S^{d-1}$, suppose $u$ is the solution to

[TABLE]

and $v$ is the solution to

[TABLE]

For any $\mathbf{R}^{d}$, for some $c_{d}$, for any $y\in\mathbf{R}^{d}$,

[TABLE]

Without loss of generality suppose $x=|x|e_{1}$. Then,

[TABLE]

which implies $(\ref{2.15.1})$. Then by theorem 2.2 and averaging over $\omega\in S^{d-1}$, taking $\partial_{\omega}f=\omega\cdot\nabla f$, and $x_{\omega}=x\cdot\omega$,

[TABLE]

Then by Hölder’s inequality and averaging over $\omega\in S^{d-1}$, for any $R>0$,

[TABLE]

Therefore, for $k\leq j$,

[TABLE]

This is a useful estimate since it opens up the possibility of integrating by parts, and moving a derivative to a more advantageous position. This opportunity will prove to be quite useful, since we will follow the analysis in [2]. That is, if $\phi$ be a solution to $(\ref{1.1})$, let

[TABLE]

Then, choosing to work in the Coulomb gauge, for $m=1,...,d$,

[TABLE]

where

[TABLE]

and

[TABLE]

3 Growth of Besov norms

Now we utilize frequency envelopes to control the growth of the Besov norm. The frequency envelopes were introduced in [18] for the study of wave maps. Frequency envelopes majorize the size of a Littlewood - Paley projection of a function, while at the same time smoothing out the differences in size at different frequency levels. Let $\delta>0$ be a small constant, say $\delta=\frac{1}{4}$, and let

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Remark: From the definition,

[TABLE]

Also observe that for any $j$,

[TABLE]

and by Young’s inequality,

[TABLE]

Now if $\langle f,g\rangle$ is the inner product

[TABLE]

then

[TABLE]

so by $(\ref{2.19})$,

[TABLE]

Now by Fourier support arguments,

[TABLE]

By a similar calculation,

[TABLE]

Now split

[TABLE]

Because $\frac{\partial_{l}\partial_{m}}{\Delta}$ is a bounded Fourier multiplier,

[TABLE]

Meanwhile, because $(P_{\leq k-20}\bar{\psi}_{l})(P_{\leq k-20}\psi_{m})$ is supported on $|\xi|\leq 2^{k-15}$, by $(\ref{2.7})$ we compute

[TABLE]

Also by $(\ref{2.7})$,

[TABLE]

Therefore, combining $(\ref{3.10})$ - $(\ref{3.14})$ with an interpolation argument proves

[TABLE]

Next, integrating by parts, since $\sum_{l=1}^{d}\partial_{l}A_{l}=0$

[TABLE]

To estimate

[TABLE]

observe that if $m$ is a Fourier multiplier satisfying $|\nabla m(\xi)|\lesssim\frac{1}{|\xi|}$ and $|m(\xi)|\lesssim 1$, then by the fundamental theorem of calculus, if $|\eta|<<|\xi|$,

[TABLE]

If $|\eta|\gtrsim|\xi|$,

[TABLE]

Therefore, by $(\ref{2.20})$

[TABLE]

may be estimated in a manner identical to the argument leading up to $(\ref{3.15})$.

It finally remains to estimate

[TABLE]

By the Sobolev embedding theorem and Bernstein’s inequality,

[TABLE]

Next, by the Sobolev embedding theorem and $(\ref{3.21})$,

[TABLE]

Therefore, by Fourier support properties,

[TABLE]

Now since the Fourier multiplier $2^{m}\frac{\partial_{k}}{\Delta}P_{m}$ is uniformly bounded, by an argument similar to $(\ref{3.11})$ - $(\ref{3.15})$,

[TABLE]

On the other hand, if $m\geq k-10$, then by Fourier support arguments and the Sobolev embedding theorem,

[TABLE]

and therefore,

[TABLE]

In conclusion, we have proved

Theorem 3.1

The frequency envelope $\alpha$ has the bounds

[TABLE]

4 Bilinear estimates

Now we use the interaction Morawetz estimates to prove some bilinear estimates.

Proposition 4.1

For any $j$,

[TABLE]

Proof: Take $(\ref{2.17})$ and set $\mathcal{N}_{1}=\mathcal{N}_{2}=\mathcal{N}$, $u=v=\psi_{x}$, where

[TABLE]

We will call the first line on the right hand side of $(\ref{2.17})$ the main term, and the other lines the error. Then we label the contribution to the error from $\mathcal{N}^{(1)}$, $\mathcal{E}^{(1)}$, and so on. The main term may be easily estimated.

[TABLE]

Computing $\mathcal{E}^{(1)}$, $(\ref{3.9})$ implies

[TABLE]

Similarly, by $(\ref{3.10})$ - $(\ref{3.16})$,

[TABLE]

and by $(\ref{3.20})$ - $(\ref{3.26})$,

[TABLE]

To estimate the integral of $\mathcal{E}^{(4)}$, split

[TABLE]

and split $\mathcal{E}^{(4)}=\mathcal{E}^{(4)}_{1}+\mathcal{E}^{(4)}_{2}$. Then by $(\ref{3.17})$ - $(\ref{3.19})$,

[TABLE]

To estimate the contribution of $\mathcal{E}^{(4)}_{1}$ we again integrate by parts. First,

[TABLE]

Then integrating by parts,

[TABLE]

[TABLE]

Now recall the definition of $A_{x}$ in $(\ref{2.20})$. Then we can split

[TABLE]

Now by definition of $\gamma_{k}$ and $\beta_{k}$, if $\delta<\frac{1}{4}$, then by the Sobolev embedding theorem,

[TABLE]

Therefore,

[TABLE]

Now by Fourier support properties,

[TABLE]

is supported on $|\xi|\leq 2^{k-5}$, so

[TABLE]

The Fourier multiplier $P_{m}(\frac{\partial_{k}}{\Delta})$ has size $\sim 2^{-m}$, so by $(\ref{3.13})$ and $(\ref{3.14})$,

[TABLE]

Also, by the Sobolev embedding theorem and Bernstein’s inequality,

[TABLE]

Therefore, by Hölder’s inequality,

[TABLE]

Also,

[TABLE]

so following the analysis in $(\ref{4.9})$ - $(\ref{4.19})$ with $l$ and $k$ swapped proves

[TABLE]

This finally proves proposition 4.1. $\Box$

Proof of theorem 1.3: Theorem 1.3 may be proved by a bootstrap argument. We have a local existence result.

Proposition 4.2

Assume $\phi_{0}\in H_{Q}^{\infty}$. Then there is $T_{\sigma_{0}}=T(\|\phi_{0}\|_{H_{Q}^{\sigma_{0}}})>0$ and a solution $\phi\in C([-T_{\sigma_{0}},T_{\sigma_{0}};H_{Q}^{\infty})$ of the initial value problem $(\ref{1.1})$. Additionally, $T_{\sigma_{0}}$ can be chosen so that

[TABLE]

and for $\sigma\in[\sigma_{0},\infty)\cap\mathbf{Z}$,

[TABLE]

Proof: See [9]. $\Box$

Then for $\phi_{0}\in H_{Q}^{\sigma_{0}}$, for $\sigma_{0}$ sufficiently large, and $T(\|\phi_{0}\|_{H_{Q}^{\sigma_{0}}})>0$ sufficiently small, we have that

[TABLE]

for $\epsilon(\|\phi_{0}\|_{\dot{H}^{d/2}})$ sufficiently small. Then by theorem 3.1 and proposition 4.1,

[TABLE]

and

[TABLE]

Again by theorem 3.1 and proposition 4.1,

[TABLE]

and

[TABLE]

Combining $(\ref{4.27})$ and $(\ref{4.28})$,

[TABLE]

But we could then replace $\alpha_{j}$ and $\gamma_{j}$ by $\tilde{\alpha}_{j}$ and $\tilde{\gamma}_{j}$, with

[TABLE]

and

[TABLE]

Following the arguments proving theorem 3.1 and proposition 4.1, it is possible to prove that

[TABLE]

and

[TABLE]

and therefore

[TABLE]

In particular, this implies that for any $t\in[-T_{\sigma_{0}},T_{\sigma_{0}}]$,

[TABLE]

Then we may extend the interval of existence a little farther, but still maintaining $(\ref{4.24})$, which then implies that $(\ref{4.35})$ holds. We can iterate this argument, and the implicit constants in $(\ref{4.29})$ and $(\ref{4.35})$ are uniformly bounded. Making a standard bootstrap argument then implies theorem 1.3. $\Box$