# Need help understanding this proof of regularity of traveling wave solutions to the Gross-Pitaevskii equation

These are actually 4 question about a proof given in this paper. Any hint to solutions for any of these questions would be much appreciated!

Lemma 1. Assume $v$ is a solution to the equation $$ic \partial_1 v + \Delta v + v(1-\vert v \vert^2) =0 \tag{GP}$$ in $L^1_{loc}(\mathbb{R}^3)$ of finite energy, then $v$ is regular, bounded and its gradient belongs to all the spaces $W^{k,p}(\mathbb{R}^3)$ for $k \in \mathbb{3}$ and $p \in [2,\infty]$.

Here "finite energy" means, that $$\frac{1}{2} \int_{\mathbb{R}^N}\vert \nabla v \vert^2+\frac{1}{4}\int_{\mathbb{R}^N}(1-\vert v \vert^2)^2 < \infty.$$

Proof. The authors consider a point $z_0 \in \mathbb{R}^3$ and denote by $\Omega$ the unit ball with center $z_0$. Then the consider the solutions $v_1$ and $v_2$ of $$\begin{cases} \Delta v_1=0 & \mbox{on } \Omega \\ v_1 = v & \mbox{on } \partial \Omega\end{cases} \tag{1}$$ and $$\begin{cases} \Delta v_2=g(v):=v(1-\vert v\vert^2) + ic \partial_1 v & \mbox{on } \Omega \\ v_2 = 0 & \mbox{on } \partial \Omega\end{cases} \tag{2}$$

Question 1. It seams odd to consider $v \in L^1_{loc}(\mathbb{R}^3)$. A function in this space can not even be a weak solution to (GP), can it? Also equation (1) does not need have a solution if $v$ is only $L^1$, does it?

The authors go on and show that $g(v)$ is uniformly bounded in $L^{\frac{4}{3}}(\Omega)$, i.e. $\Vert g(v) \Vert_{L^{\frac{4}{3}}(\Omega)}$ is bounded by a constant not depending on $z_0$. The infer that $v_1$ is unformly bounded in $L^4(\Omega)$ and $v_2$ in $W^{2,\frac{4}{3}}(\Omega)$. I do already understand this part (some elliptic arguments and Sobolev embeddings).

But now they denote by $\omega$ the ball with center $z_0$ and radius $\frac{1}{2}$ and tell me to use Cacioppoli inequalities to show that $v_1$ is uniformly bounded on $W^{2,\frac{4}{3}}(\omega)$ and $W^{3,\frac{12}{11}}(\omega)$. This would show that $v$ is uniformly bounded in $W^{2,\frac{4}{3}}(\Omega)$.

Question 2. How do you do that? By Caccioppoli inequalities for the Laplace equation I can show that $\Vert Dv_1 \Vert_{L^2(\omega)} \Vert \leq C \Vert v_1 \Vert_{L^2(\omega)}$ but nothing else. How do they do this? Why do they even need the second statement (i.e. $W^{3,\frac{12}{11}}$)?

They go on and compute $$\nabla g(v) = \nabla v(1-\vert v\vert^2) - 2 (v.\nabla v)v + ic \partial_1 \nabla v$$ which is easy. But from this they infer that $g(v)$ is uniformly bounded in $L^\frac{12}{11}(\omega)$ and finally $v$ is uniformly bounded in $C^{0,\frac{1}{12}}(\omega)$.

Question 3. I don't understand any of these steps. For the first one I try to estimate $\int \vert \nabla v - \nabla v \vert v \vert^2 \vert^\frac{12}{11} \leq \int \vert \nabla v \vert^\frac{12}{11} + \int \vert \nabla v \vert^\frac{12}{11} \vert v \vert^\frac{24}{11}$ which I can't manage. For the second one I don't understand why they pick the $\frac{1}{12}$-Hölder space of all Hölder spaces.

Last but not least they set $$h(w)=w(1-\vert v \vert^2) - 2 (v.w)v + \left(\frac{c^2}{2}+2\right)w$$ where $w=\nabla v$ and state that $h(w)$ belongs to $L^2(\mathbb{R}^N)$ and $w$ to $H^2(\mathbb{R}^N)$.

Question 4. Why is $\nabla v$ even differentiable enough to be plugged in $h$? How do they infer their two results? I have absolutely no idea.

Thank you so much in advance!

Question 1. The function is "in $$L^1_{\rm loc}(\mathbb R^3)$$ of finite energy". The part "in $$L^1_{\rm loc}(\mathbb R^3)$$" is actually redundant here, since the finiteness of energy gives $$L^4_{\rm loc}$$ directly (and $$L^6_{\rm loc}$$ via Sobolev embedding of $$W^{1,2}$$ in three dimensions). Also, since $$v\in W^{1,2}$$, the Dirichlet problem (1) for $$v_1$$ makes sense: the boundary condition $$v_1=v$$ is understood as $$v_1-v\in W^{1,2}_0(\Omega)$$. Observe that since $$v_1$$ minimizes Dirichlet energy subject to this boundary condition, we have $$\int_{\Omega}|\nabla v_1|^2\le \int_{\Omega}|\nabla v|^2\le 2E(v) \tag{A}$$

The weak form of (GP) is $$\int \left\{ - ic v\phi_1 - \nabla v\cdot \nabla \phi + v(1-\vert v \vert^2)\phi\right\} =0\tag{WGP}$$ for all smooth compactly supported functions $$\phi$$. This makes sense as long as $$\nabla v$$ and $$v^3$$ are integrable, which again follows from the finiteness of energy.

Question 2. I don't know what form of Caccioppoli's inequality the author had in mind, but one does not need Caccioppoli to estimate the derivatives of a harmonic function. Indeed, $$|\nabla v_1|^2$$ is a subharmonic function, which means its value at any point is bounded by the average over a ball centered at that point. If $$x\in \frac{3}{4}\Omega$$ (ball of radius $$3/4$$), then $$B(x,1/4)\subset \Omega$$, hence $$|\nabla v_1(x)|^2\le \frac{1}{\pi/16}\int_{\Omega}|\nabla v_1|^2$$, where the integral is controlled by (A). This gives a uniform upper bound on the values of $$\nabla v_1$$ in $$\frac{3}{4}\Omega$$. Since $$\nabla v_1$$ is harmonic, the interior regularity for harmonic equations (Theorem 2.10 in Gilbarg-Trudinger) gives you control over all derivatives of $$v_1$$ in $$\omega$$ in terms of the supremum of $$|\nabla v_1|$$ on the boundary of the ball $$\frac{3}{4}\Omega$$. You have uniform boundedness in any Sobolev space you want.

The part "so, $$v$$ is uniformly bounded in $$W^{2;4/3}(\omega)$$" follows by recalling that $$v=v_1+v_2$$ and $$v_2$$ is uniformly bounded in $$W^{2;4/3}(\omega)$$.

Question 3. Recall that $$v\in L^6(\omega)$$ with a uniform bound coming from the Sobolev embedding. The formula for $$\nabla g(v)$$ has things like $$|\nabla v| v^2$$, which is the product of $$L^2$$ function with $$L^3$$ function. By Hölder's inequality, such a product is in $$L^p$$ with $$\frac{1}{p}=\frac{1}{2}+\frac{1}{3}$$, hence $$p=6/5$$. There's also $$\partial_1 \nabla v$$ which is in $$L^{4/3}$$ because $$v\in W^{2,4/3}$$. All in all, $$\nabla g\in L^{6/5}$$. This places $$v_2$$ into $$W^{3,6/5}$$ from where it lands (via Morrey-Sobolev) into $$C^{0,1/2}$$. The numerology is $$\frac{1}{\text{integrability you have}} - \frac{\text{derivatives you give up}}{\text{dimension}} = \begin{cases} 1/q \\ -\alpha/\text{dimension} \end{cases}$$ where $$q$$ is the new integrability exponent (if the result is positive) and $$\alpha$$ is the Hölder exponent (if the result is negative).

I don't know why the author has $$12/11$$ and $$1/12$$ for the Sobolev and Hölder exponents, they are worse than what I got above.

Question 4. Since $$v$$ is in $$W^{3,6/5}$$, the equation (GP) not only holds in the strong sense, but leaves the room for differentiating it once. The expression $$h(w)$$ is misleadingly named since it obviously depends on $$v$$ as well. Anyway, $$w\in L^2$$ because $$u$$ has finite energy and multiplication by something involving $$v$$ is harmless ($$v$$ is bounded). This is why $$h(w)$$ is in $$L^2$$.

The reason why the author added $$(c^2/2+2)w$$ to both sides is to have a uniformly elliptic operator $$L$$ on the left side of $$Lw=h(w)$$. The symbol of $$L$$ is $$\xi_1^2+\xi_2^2+\xi_3^2+c\xi_1+(c^2/2+2)$$ which is appropriately positive in $$\mathbb R^3$$. Ellipticity and $$Lw\in L^2$$ give $$w\in H^2$$.

• Thank you so much for all our effort! This is really helpful!
– mjb
Aug 14, 2013 at 8:05
• One more question, if I may: In your answer to question 1 you infer $v \in L^4_{loc}$, which is fine, but then you say $v \in W^{1,2}$ which would imply that $v \in L^2$. Is that correct? Or did you mean $loc$? In fact we know by the finiteness of the Energy that $1-\vert v \vert^2 \in L^2$. But then $\int 1 - 2\vert v \vert^2+\vert v \vert^4 =: \int A-B+C < \infty$. Since $\int A+C = \infty$ cleary $\int B =\infty$ and therefore $v \not\in L^2$, no?
– mjb
Aug 14, 2013 at 12:09
• @mjb I did not attach any domain to $W^{1,2}$ but I meant $\Omega$, not $\mathbb R^3$. Aug 14, 2013 at 17:37
• Ah of course! Thank you so much!
– mjb
Aug 15, 2013 at 7:46
• @mjb A function harmonic in any sensible way is classically harmonic, by Weyl's lemma. Aug 15, 2013 at 15:58