Weak Universality of the Dynamical ${{\Phi }_{3}^{4}}$ Model on the Whole Space

Abstract

We prove that the large scale behavior of a class of stochastic weakly nonlinear reaction-diffusion models on $\mathbb {R}^{3}$ converges to the dynamical ${{\Phi }^{4}_{3}}$ model using paracontrolled distributions on weighted Besov spaces. Our approach depends on the delicate choice of the weight, the localization operator technique and a modified version of the maximum principle from Gubinelli and Hofmanová (Commun. Math. Phys. 368, 1201–1266, 5).

Appendices

Appendix A: Extra Estimates

A.1 Extra Estimates for ϕ in $C\mathcal {C}^{\alpha }(\rho ^{\frac {3+6\alpha }{2m}})$

The calculations in this section is similar as in [5]. Only the weights are more delicate. The idea is to use Lemma 2.2 to control terms containing ${\mathscr{V}}_>$. Then by paraproduct estimate the result follows. For the terms in Φ omitted in Section ?? we use Lemma 2.2 to choose L in the localization operator to have

which combined with paraproduct estimate Lemma 2.5 imply the following estimates:

2.1 A.2 Extra Estimates for ϕ in $C\mathcal {C}^{\frac {1}{2}+\alpha }(\rho ^{\frac {3+6\alpha }{2m}})$

In this subsection we consider the extra terms which we omit in Section ??. Similar as in Section ?? we have to use $\|\psi \|_{L^{\infty } L^{\infty }(\rho ^{\frac {1}{2}+\alpha })}$ to control various power of 2^L. By the localization operator chosen in Section A and Lemma 2.2 we have

which combined with Lemma 2.5 and (??), (??) yield the following estimates:

2.2 A.3 Extra Estimates for 𝜗

In this subsection we consider the extra terms which we omit in Section ??. By the localization operator chosen in Section A and Lemma 2.2 we have

which combined with Lemma 2.5 and (??), (??), (??) yield

2.3 A.4 Extra Bounds for ψ in $C\mathcal {C}^{2-\gamma }(\rho ^{\frac {3}{2}+\gamma _{1}})$

In this section we deduce the extra bounds for Section ??. Besides Lemmas 2.2 and 2.5 we also need to use commutator estimate in Lemmas 2.6 and 2.7. We first concentrate on the terms not containing localization operators. By (??) and Lemma 2.2 we have

Using Lemma 2.5 and Lemma 2.7, (??) and Lemma 2.1 we deduce

By Lemma 2.6 and (??) we have

Lemma 2.5 and (??) imply that

By Lemma 2.5 and (??) we find

Using (??) we obtain

Using paraproduct estimates Lemma 2.5 and (??)–(??) we deduce

Furthermore, by using Lemma 2.2 we have

which combined with Lemma 2.5, (??) and γ₁ > 4α give

Moreover, we also use Lemma 2.2 to have

which combined with Lemma 2.5 and (??) imply

We also have

provided .

Appendix B:: Global Well-Posedness for Smooth Noise Case

Proposition B.1

Let T > 0,C₀ > 0, $\eta \in C^{\infty }([0,T]\times \mathbb {T}^{d})$, $\varphi _{0}\in C^{\infty }(\mathbb {T}^{d})$ and G as given in the introduction. There exists a unique classical solution $\varphi \in L_{T}^{\infty } \mathcal {C}_{T}^{2+\alpha }(\mathbb {T}^{d})\cap C^{1}L^{\infty }(\mathbb {T}^{d})$ to

$$ \mathcal{L} \varphi + C_{0}\varphi^{m} =-G(\varphi) +\eta,\qquad\varphi(0)=\varphi_{0}. $$

(B.1)

Proof

The existence and uniqueness of weak solutions can be obtained by monotonicity argument. We then prove a priori estimate in L^p, p > 1. We test by φ^2p− 1 to obtain

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{2p}\partial_{t}{\int}_{\mathbb{T}^{d}} |\varphi|^{2p}dx+(2p-1){\int}_{\mathbb{T}^{d}} |\varphi|^{2p-2}|\nabla\varphi|^{2} dx+C_{0}{\int}_{\mathbb{T}^{d}} |\varphi|^{2p+m-1}dx \\&\leq&{\int}_{\mathbb{T}^{d}} |G(\varphi)||\varphi|^{2p-1}dx+{\int}_{\mathbb{T}^{d}} |\eta||\varphi|^{2p-1}dx. \end{array} $$

By our assumption on G(φ) we know that for κ > 0 |G(φ)|≤C + (C₀ −κ)|φ|^m, which implies the right hand side of the above inequality can be controlled by

$$ (C_{0}-\kappa){\int}_{\mathbb{T}^{d}} |\varphi|^{2p+m-1}dx+C{\int}_{\mathbb{T}^{d}} |\varphi|^{2p-1}dx. $$

Then we have

$$ \frac{1}{2p}\partial_{t}{\int}_{\mathbb{T}^{d}} |\varphi|^{2p}dx+(2p-1){\int}_{\mathbb{T}^{d}} |\varphi|^{2p-2}|\nabla\varphi|^{2} dx+C_{0}{\int}_{\mathbb{T}^{d}} |\varphi|^{2p+m-1}dx\leq C_{T,p}. $$

By using Sobolev embedding we know that $\varphi , \varphi ^{m}, G(\varphi )\in L_{T}^{\infty } \mathcal {C}^{-\alpha }(\mathbb {T}^{d})$ for α > 0 and small enough. Then using Lemma 2.3 on the periodic setting, we know that $\varphi \in L_{T}^{\infty } \mathcal {C}^{2-\alpha }(\mathbb {T}^{d})\cap C_{T}^{(2-\alpha )/2}L^{\infty }(\mathbb {T}^{d})$. Then by simple calculation we obtain that $\varphi ^{m}, G(\varphi )\in L_{T}^{\infty } \mathcal {C}^{\alpha }(\mathbb {T}^{d})$, which implies that $\varphi \in L_{T}^{\infty } \mathcal {C}_{T}^{2+\alpha }(\mathbb {T}^{d})\cap C^{1}L^{\infty }(\mathbb {T}^{d})$ is a classical solution to (B.1). □

Proposition B.2

Let T > 0,C₀ > 0 and let ρ be a polynomial weight, $\eta \in C_{T}\mathcal {C}^{\gamma }(\rho ^{\frac {3}{2}+\gamma _{1}})\cap L_{T}^{\infty } L^{\infty }(\rho ^{\frac {3+6\alpha }{2}})$ and $\varphi _{0}\in \mathcal {C}^{2+\gamma }(\rho ^{\frac {3}{2}+\gamma _{1}})\cap L^{\infty }(\rho ^{\frac {3+6\alpha }{2m}})$. There exists a unique classical solution $\varphi \in C_{T}\mathcal {C}^{2+\gamma }(\rho ^{\frac {3}{2}+\gamma _{1}})\cap {C_{T}^{1}}L^{\infty }(\rho ^{\frac {3}{2}+\gamma _{1}})\cap L_{T}^{\infty } L^{\infty } (\rho ^{\frac {3+6\alpha }{2m}})$ to

$$ \mathcal{L} \varphi + C_{0}\varphi^{m} =-G(\varphi) +\eta,\qquad\varphi(0)=\varphi_{0}. $$

(B.2)

Proof

Consider the periodization η_M and ${\varphi _{0}^{M}}$ on $\mathbb {T}^{d}_{M}$. By Proposition B.1 there exists a classical solution φ_M to (B.2) with η,φ₀ replaced by η_M and ${\varphi _{0}^{M}}$, respectively. In the following we obtain the uniform estimates for φ_M. Since the estimate is independent of M, we omit M for simplicity. By similar argument as in the proof of Lemma 2.4 we have

$$ C_{0}\|\varphi\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m} \leq C_{0}\|\varphi_{0}\|_{L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m}+ \|G(\varphi)\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}+\|\eta\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}. $$

By Assumption 1 in introduction we know that

$$ \|G(\varphi)\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}\leq C_{\delta}+(\frac{C_{1}}{m_{1}!}+\delta)\|\varphi\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m_{1}}\leq C_{\delta}+(C_{0}-\delta)\|\varphi\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m}, $$

where for m₁ < m we can use Young’s inequality and for m₁ = m we use (??) in the last inequality. Now we have

$$ \|\varphi\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m}\lesssim\|\varphi_{0}\|_{L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m} +\|\eta\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}. $$

Moreover, we have

$$ \begin{array}{@{}rcl@{}} &&\|G(\varphi)\|_{C_{T}\mathcal{C}^{\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})} + \| \varphi^{m}\|_{C_{T}\mathcal{C}^{\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})}\lesssim (1 + \|\varphi\|_{L^{\infty}_{T} L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m-1})\|\varphi\|_{C_{T}\mathcal{C}^{\gamma}(\rho^{\frac{3+6\alpha}{2m}+\gamma_{1}-3\alpha})}\\ &\lesssim& (1+\|\varphi\|_{L_{T}^{\infty} L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m-\frac{\gamma}{2+\gamma}}) \|\varphi\|_{C_{T}\mathcal{C}^{2+\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})}^{\frac{\gamma}{2+\gamma}} \\&\lesssim& \|\varphi_{0}\|_{L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m+1} +\|\eta\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}^{2}+1+\delta_{0}\|\varphi\|_{C_{T}\mathcal{C}^{2+\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})}, \end{array} $$

for 0 < δ₀ < 1. Now we have the following uniform estimate by Lemma 2.3

$$ \begin{array}{@{}rcl@{}} &&\| \varphi\|_{C_{T}\mathcal{C}^{2+\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})}+\| \varphi\|_{{C_{T}^{1}}L^{\infty}(\rho^{\frac{3}{2}+\gamma_{1}})} \\&\lesssim& \|\varphi_{0}\|_{L^{\infty}(\rho^{\frac{3+6\alpha}{2m}})}^{m+1}+\|\varphi_{0}\|_{\mathcal{C}^{2+\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})} +\|\eta\|_{L_{T}^{\infty}L^{\infty}(\rho^{\frac{3+6\alpha}{2}})}^{2}+\|\eta\|_{C_{T}\mathcal{C}^{\gamma}(\rho^{\frac{3}{2}+\gamma_{1}})}+1. \end{array} $$

Since the constant we omit in the above estimate is independent of M, we can obtain compactness of the approximate sequence φ_M in a slightly worse space, which allows to pass to the limit in the approximate equation. We can also obtain the solution belong to the spaces where the uniform bounds hold. For the uniqueness in the above weighted space we can also choose time dependent weight $\pi (t,x)=\exp (-t\rho ^{-2b}(x))$ for ρ = 〈x〉^− 1 and b ∈(0,1/2) as in [5]. Take two different solutions φ₁ and φ₂ starting from the same initial data φ₀ and satisfying the above bounds. Set u := φ₁ −φ₂. Then we have

$$ \mathcal{L} u + C_{0}({\varphi_{1}^{m}}-{\varphi_{2}^{m}}) =-G(\varphi_{1})+G(\varphi_{2}). $$

Now we take inner product with π²u in L² and use ∂_tπ = −πρ^− 2b to have

$$ \frac{1}{2}\partial_{t}\|u\|_{L^{2}(\pi)}^{2}+\|\nabla u\|_{L^{2}(\pi)}^{2} +\|\rho^{-b}u\|_{L^{2}(\pi)}^{2} \!\leq \!\langle-G(\varphi_{1})+G(\varphi_{2}),u\pi^{2}\rangle+C_{\delta} \|u\|_{L^{2}(\pi)}^{2}+\delta\|\nabla u\|_{L^{2}(\pi)}^{2}, $$

where we use $|\frac {\nabla \pi }{\pi }|\lesssim 1$.

$$ \begin{array}{@{}rcl@{}} |\langle-G(\varphi_{1})+G(\varphi_{2}),u\pi^{2}\rangle|&\lesssim & (1+\|\varphi_{1}\|_{L^{\infty}(\rho^{\frac{b}{m_{1}-1}})}^{m_{1}-1}+\|\varphi_{2}\|_{L^{\infty}(\rho^{\frac{b}{m_{1}-1}})}^{m_{1}-1})\|\rho^{-\frac{b}{2}}u\|_{L^{2}(\pi)}^{2} \\&\lesssim & \delta\|\rho^{-b}u\|_{L^{2}(\pi)}^{2}+C_{\delta}\|u\|_{L^{2}(\pi)}^{2}. \end{array} $$

Now the uniqueness follows by Gronwall’s Lemma. □