$L_p$-solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

Abstract

This paper establishes $L_p$-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise:

$$\begin{aligned} \partial _t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\bar{b}}^i u u_{x^i} + \partial _t^\beta \int _0^t \sigma (u)dW_t,\quad t>0;\quad u(0,\cdot ) = u_0, \end{aligned}$$

where $\alpha \in (0,1)$, $\beta < 3\alpha /4+1/2$, and $d< 4--2(2\beta -1)_+/\alpha$. The operators $\partial _t^\alpha$ and $\partial _t^\beta$ are the Caputo fractional derivatives of order $\alpha$ and $\beta$, respectively. The process $W_t$ is an $L_2(\mathbb {R}^d)$-valued cylindrical Wiener process, and the coefficients $a^{ij}, b^i, c, {\bar{b}}^{i}$ and $\sigma (u)$ are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant $T<\infty$, small $\varepsilon >0$, and almost sure $\omega \in \varOmega$,

$$\begin{aligned} \sup _{x\in \mathbb {R}^d}|u(\omega ,\cdot ,x)|_{C^{\left[ \frac{\alpha }{2}\left( \left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 \right) +\frac{(2\beta -1)_{-}}{2} \right] \wedge 1-\varepsilon }([0,T])}<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{t\le T}|u(\omega ,t,\cdot )|_{C^{\left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 - \varepsilon }(\mathbb {R}^d)} < \infty . \end{aligned}$$

The Hölder regularity of the solution in time changes behavior at $\beta = 1/2$. Furthermore, if $\beta \ge 1/2$, then the Hölder regularity of the solution in time is $\alpha /2$ times that in space.

Appendix

This section presents the published results of Bessel potential spaces and fractional calculus for the convenience of the reader.

1.1 Properties of Bessel potential spaces

Below we present the properties of Bessel potential spaces $H_{p}^{\gamma }$. For the definition of the Bessel potential spaces $H_{p}^{\gamma }$, see Definition 2.4.

Next, we introduce the space of point-wise multipliers in $H_p^\gamma$.

Definition A.1

Fix $\gamma \in \mathbb {R}$ and $\alpha \in [0,1)$ such that $\alpha = 0$ if $\gamma \in \mathbb {Z}$ and $\alpha >0$ if $|\gamma |+\alpha$ is not an integer. Define

$$\begin{aligned}{} & {} \begin{aligned} B^{|\gamma |+\alpha } = {\left\{ \begin{array}{ll} B(\mathbb {R}^{d}) &{}\quad \text {if } \gamma = 0, \\ C^{|\gamma |-1,1}(\mathbb {R}^{d}) &{}\quad \text {if }\gamma \text {is a nonzero integer}, \\ C^{|\gamma |+\alpha }(\mathbb {R}^{d}) &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned} \\{} & {} \begin{aligned} B^{|\gamma |+\alpha }(\ell _2) = {\left\{ \begin{array}{ll} B(\mathbb {R}^{d},\ell _2) &{}\quad \text {if } \gamma = 0, \\ C^{|\gamma |-1,1}(\mathbb {R}^{d},\ell _2) &{}\quad \text {if} \gamma \text {is a nonzero integer}, \\ C^{|\gamma |+\alpha }(\mathbb {R}^{d},\ell _2) &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned} \end{aligned}$$

where $B(\mathbb {R}^{d})$ is the space of bounded Borel functions on $\mathbb {R}^{d}$, $C^{|\gamma |-1,1}(\mathbb {R}^{d})$ represents the space of $|\gamma |-1$ times continuous differentiable functions whose derivatives of the $(|\gamma |-1)$th order derivative are Lipschitz continuous, and $C^{|\gamma |+\alpha }$ is the real-valued Hölder spaces. The space $B(\ell _2)$ represents a function space with $\ell _2$-valued functions instead of real-valued function spaces.

Lemma A.1

Let $\gamma \in \mathbb {R}$ and $p>1$.

• (i) The space $C_c^\infty (\mathbb {R}^d)$ is dense in $H_{p}^{\gamma }$.

• (ii) Let $\gamma - d/p = n+\nu$ for some $n=0,1,\cdots$ and $\nu \in (0,1]$. Then, for any $k\in \{ 0,1,\cdots ,n \}$, we obtain

  $$\begin{aligned} | D^k u |_{C(\mathbb {R}^d)} + | D^n u |_{\mathcal {C}^\nu (\mathbb {R}^d)} \le N \Vert u \Vert _{H_{p}^\gamma }, \end{aligned}$$

  (A.1)

  where $\mathcal {C}^\nu (\mathbb {R}^d)$ is the Zygmund space.

• (iii) The operator $D_i:H_p^{\gamma }\rightarrow H_p^{\gamma +1}$ is bounded. Moreover, for any $u\in H_p^{\gamma +1}$,

  $$\begin{aligned} \left\| D^i u \right\| _{H_p^\gamma } \le N\Vert u \Vert _{H_p^{\gamma +1}}, \end{aligned}$$

  where $N = N(\gamma ,p)$.

• (iv) For $\gamma \in (0,1)$ and $u\in H_{p}^{\gamma }$,

  $$\begin{aligned} \Vert (1- \varDelta ^{\gamma })u \Vert _{L_p} \le N\left( \Vert u \Vert _{L_p} + \Vert (-\varDelta )^{\gamma } u \Vert _{L_p} \right) , \end{aligned}$$

  where $N = N(\gamma ,p)$ and $(-\varDelta )^{\gamma /2}$ is the fractional Laplacian.

• (v) For any $\mu ,\gamma \in \mathbb {R}$, the operator $(1-\varDelta )^{\mu /2}:H_p^\gamma \rightarrow H_p^{\gamma -\mu }$ is an isometry.

• (vi) Let

  $$\begin{aligned} \begin{aligned} \varepsilon \in [0,1],\quad p_i\in (1,\infty ),\quad \gamma _i\in \mathbb {R},\quad i=0,1,\\ \gamma =\varepsilon \gamma _1+(1-\varepsilon )\gamma _0,\quad 1/p=\varepsilon /p_1+(1-\varepsilon )/p_0. \end{aligned} \end{aligned}$$

  Then, we have

  $$\begin{aligned} \Vert u\Vert _{H^\gamma _{p}} \le \Vert u\Vert ^{\varepsilon }_{H^{\gamma _1}_{p_1}}\Vert u\Vert ^{1-\varepsilon }_{H^{\gamma _0}_{p_0}}. \end{aligned}$$

• (vii) Let $u\in H_p^\gamma$. Then, we have

  $$\begin{aligned} \Vert au \Vert _{H_p^\gamma } \le N\Vert a \Vert _{B^{|\gamma |+\alpha }}\Vert u \Vert _{H_p^\gamma }\quad \text {and}\quad \Vert bu \Vert _{H_p^\gamma (\ell _2)} \le N\Vert b \Vert _{B^{|\gamma |+\alpha }(\ell _2)}\Vert u \Vert _{H_p^\gamma }, \end{aligned}$$

  where $N = N(\gamma ,p)$. The spaces $B^{|\gamma |+\alpha }$ and $B^{|\gamma |+\alpha }(\ell _2)$ are introduced in Definition A.1.

Proof

The above results are well known. For (i), (iii), (v), and (vi), see Theorems 13.3.7 (i), 13.8.1, 13.3.7 (ii), and Exercise 13.3.20 of [28], respectively. For (ii), see [34]. For (iv), see Theorems 1.3.6 and 1.3.8 of [11]. For (vii), the reader is referred to [26, Lemma 5.2]. $\square$

Next, we propose embedding theorems for Slobodetskii’s spaces (e.g., [34]).

Lemma A.2

If $\mu p >1$ and $p\ge 1$, for any continuous $L_p$-valued function $\phi$ and $\gamma \le \rho$, we obtain the following:

$$\begin{aligned} \begin{aligned}&\Vert \phi (\rho ) - \phi (\gamma ) \Vert _{L_p}^p \\&\quad \le N(\rho -\gamma )^{\mu p -1}\int _\gamma ^\rho \int _\gamma ^\rho 1_{t > s}\frac{\Vert \phi (t) - \phi (s) \Vert _{L_p}^p}{|t-s|^{1+\mu p}} dsdt\quad \left( \frac{0}{0}:= 0 \right) , \end{aligned} \end{aligned}$$

(A.2)

where $N = N(\mu ,p)$. In particular,

$$\begin{aligned} \begin{aligned} \mathbb {E}\sup _{0\le s < t \le T}\frac{\Vert \phi (t) - \phi (s) \Vert _{L_p}^p}{|t-s|^{\mu p - 1}}&\le N \int _0^{T}\int _0^{T} 1_{t>s} \frac{\mathbb {E}\left\| \phi (t) - \phi (s) \right\| _{L_p}^p}{|t-s|^{1 + \mu p}}dsdt. \end{aligned} \end{aligned}$$

(A.3)

1.2 Properties of fractional calculus

We present the properties of fractional calculus. For the definitions of fractional integral $I^{\alpha }_{t}$ and derivatives $D_{t}^{\alpha }$, $\partial _{t}^{\alpha }$, see Definitions 2.2 and 2.3.

Remark A.1

• (i) For any $\alpha ,\beta \ge 0$, $D^\alpha _t D^\beta _t\varphi = D^{\alpha +\beta }_t\varphi$ and

  $$\begin{aligned} D^\alpha _t I^\beta _t \varphi = D^{\alpha -\beta }_t \varphi 1_{\alpha > \beta } + I_t^{\beta -\alpha }\varphi 1_{\alpha \le \beta }. \end{aligned}$$

  Additionally, if $\alpha \in (0,1)$, $I^{1-\alpha }_t \varphi$ is absolutely continuous, and $I^{1-\alpha }_t\varphi (0) = 0$, then the following equality holds:

  $$\begin{aligned} I^\alpha _t D^\alpha _t\varphi (t) = \varphi (t). \end{aligned}$$

• (ii) By the definition of fractional derivatives, if $\varphi (0) = \varphi ^{(1)}(0) = \cdots = \varphi ^{(n -1)}(0) = 0$, then $D^\alpha _t\varphi = \partial _t^\alpha \varphi$.

Remark A.2

For any $q \in [1,\infty ]$, by Jensen’s inequality

$$\begin{aligned} \Vert I^\alpha \varphi \Vert _{L_q((0,T))}\le N(\alpha ,p,T)\Vert \varphi \Vert _{L_q((0,T))}. \end{aligned}$$

(A.4)

Therefore, $I_t^\alpha \varphi (t)$ is well-defined and finite for almost all $t\le T$. Additionally, Fubini’s theorem implies that, for $\alpha ,\beta \ge 0$, we obtain

$$\begin{aligned} I^{\alpha +\beta }\varphi (t)=I^{\alpha }I^{\beta }\varphi (t). \end{aligned}$$

Lemma A.3 presents the relationship between the stochastic and fractional integrals, which is applied when $I_t^\alpha$ or $D_t^\alpha$ is applied to the stochastic part of the SPDEs.

Lemma A.3

Let $T<\infty$ be a constant.

• (i) Let $\alpha \ge 0$ and $h\in L_2(\varOmega \times [0,T],\mathcal {P};l_2)$. Then, the equality

  $$\begin{aligned} I^\alpha \left( \sum _{k=1}^\infty \int _0^\cdot h^k(s) dw_s^k \right) (t) = \sum _{k=1}^\infty \left( I^\alpha \int _0^\cdot h^k(s)dw_s^k \right) (t) \end{aligned}$$

  holds for all $t\le T$ almost surely and in $L_2(\varOmega \times [0,T])$, where the series on both sides converge in probability.

• (ii) If $\alpha \ge 0$ and $h_n\rightarrow h$ in $L_2(\varOmega \times [0,T],\mathcal {P};l_2)$ as $n\rightarrow \infty$, then

  $$\begin{aligned} \sum _{k = 1}^\infty \left( I^\alpha \int _0^\cdot h_n^k dw_s^k \right) (t) \rightarrow \sum _{k=1}^\infty \left( I^\alpha \int _0^\cdot h^k dw_s^k \right) (t) \end{aligned}$$

  in probability uniformly on [0, T].

• (iii) If $\alpha > 1/2$ and $h\in L_2(\varOmega \times [0,T],\mathcal {P};l_2)$, then $\left( I^\alpha \sum _{k=1}^\infty \int _0^\cdot h^k (s) dw_s^k \right) (t)$ is differentiable in t and

  $$\begin{aligned} \frac{\partial }{\partial t}\left( I^\alpha \sum _{k=1}^\infty \int _0^\cdot h^k (s) dw_s^k \right) (t) = \frac{1}{\Gamma (\alpha )}\sum _{k=1}^\infty \int _0^t (t-s)^{\alpha -1}h^k(s)dw_s^k \end{aligned}$$

  almost everywhere on $\varOmega \times [0,T]$.

Proof

Refer to Lemmas 3.1 and 3.3 of [5]. $\square$

The next lemma provides an inequality that acts like the chain rule. Although the inequality is included in the proof of [6, Proposition 4.1], we provide it for the readers’ convenience.

Lemma A.4

If $\alpha \in (0,1)$, for any $\psi \in C_c^\infty ((0,\infty )\times \mathbb {R}^d)$, we have

$$\begin{aligned} \partial _t^\alpha (\psi (\cdot ,x))^{2}(t) \le 2 \psi (t,x)|\psi (t,x)| \partial _t^\alpha \psi (t,x), \end{aligned}$$

(A.5)

for all $(t,x)\in (0,\infty )\times \mathbb {R}^d$.

Proof

Let $\psi \in C_c^\infty ((0,\infty )\times \mathbb {R}^d)$ and $t\in (0,\infty )$ and $x\in \mathbb {R}^d$. For $s\in (0,t]$, we set

$$\begin{aligned} F_1(s):= \frac{1}{2}|\psi (s,x)|^2,\quad F_2(s):= \psi (s,x)\psi (t,x), \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} F(s)&:= \frac{1}{2}\left( |\psi (s,x)|^2 - |\psi (t,x)|^2 \right) - (\psi (s,x) - \psi (t,x))\psi (t,x). \\ \end{aligned} \end{aligned}$$

Further,

$$\begin{aligned} F(s) = \frac{1}{2}|\psi (s,x) - \psi (t,x)|^2 \ge 0 \end{aligned}$$

on $s\le t$, and the equality holds for $s = t$. The integration by parts implies that

$$\begin{aligned} \begin{aligned} \int _0^t (t-s)^{-\alpha }(F_1'(s) - F_2'(s))ds = \int _0^t (t-s)^{-\alpha } F'(s)ds\le 0. \end{aligned} \end{aligned}$$

Then, by the definition of $\partial _t^\alpha$ (Definition 2.3), we obtain (5.15), with $k = 1$. $\square$

Lemma A.1 is a simplified version of [2, Theorem 8], which is a Grönwall inequality with time fractional integrations.

Theorem A.1

Let $\psi (t)$ be a nonnegative integrable function on [0, T]. For a constant $N_1$, if the function $\psi$ satisfies

$$\begin{aligned} \psi (t) \le \psi _0 + N_1I_t^\alpha \psi \end{aligned}$$

on $t\in [0,T]$, then

$$\begin{aligned} \psi (t) \le \left( 1 + \sum _{k=0}^\infty \frac{N_1^k }{\Gamma (k\alpha )}\frac{(\Gamma (\alpha )t^{\alpha })^k}{k\alpha } \right) \psi _0 \end{aligned}$$

on $t\in [0,T]$.

Proof

See [2, Theorem 8] $\square$

We discuss some facts related to the fundamental solution of fractional diffusion equations. For more information, see [24, Section 3], [25, Section 3], and [22].

Lemma A.5

Let $d\in \mathbb {N}$, $\alpha \in (0,1)$, $\beta < \alpha + 1/2$, $\gamma \in [0,2)$, and $\sigma \in \mathbb {R}$.

• (i) There exists a fundamental solution $p(t,\cdot )\in L_1(\mathbb {R}^{d})$ satisfying

  $$\begin{aligned} \partial _t^\alpha u (t,x) = \varDelta u(t,x);\quad t > 0,\quad u(0,\cdot ) = \delta _0, \end{aligned}$$

  where $\delta _0$ is the Dirac delta distribution. Furthermore, if we define $q_{\alpha ,\beta }(t,x)$ as in (7.2), for all $t\ne 0$ and $x\ne 0$,

  $$\begin{aligned} \partial _t p(t,x) = \varDelta q_{\alpha ,1}(t,x). \end{aligned}$$

  Additionally, for each $x\ne 0$, $\frac{\partial }{\partial t}p(t,x)\rightarrow 0$ as $t \downarrow 0$. Moreover, $\frac{\partial }{\partial t}p(t,x)$ is integrable in $x\in \mathbb {R}^d$ uniformly on $t\in [\varepsilon ,T]$ for any $\varepsilon >0$.

• (ii) There exist constants $c = c(\alpha ,d)$ and $N = N(\alpha ,d)$ such that if $|x|^2 \ge t^\alpha$,

  $$\begin{aligned} |p(t,x)| \le N|x|^{-d}\exp \left\{ -c|x|^{\frac{2}{2-\alpha }}t^{-\frac{\alpha }{2-\alpha }} \right\} . \end{aligned}$$

• (iii) Let $n\in \mathbb {N}$. Then, there exists $N = N(\alpha ,\gamma ,n)$ such that

  $$\begin{aligned} \left| D_t^\sigma D_x^n (-\varDelta )^{\gamma /2}q_{\alpha ,\beta }(1,x)\right| \le N\left( |x|^{-d+2-\gamma -n}\wedge |x|^{-d-\gamma -n} \right) , \end{aligned}$$

  where $q_{\alpha ,\beta }$ is the function in (7.2) and $(-\varDelta )^{\gamma /2}$ is the fractional Laplacian.

• (iv) The scaling properties hold. In other words,

  $$\begin{aligned} (-\varDelta )^{\gamma /2}q_{\alpha ,\beta }(t,x) = t^{-\frac{\alpha (d+\gamma )}{2}+\alpha -\beta } (-\varDelta )^{\gamma /2}q_{\alpha ,\beta }(1,xt^{-\frac{\alpha }{2}}), \end{aligned}$$

  (A.6)

  where $q_{\alpha ,\beta }$ is the function introduced in (7.2).

Proof

To demonstrate (i), (ii), and (iii), we follow from Theorems 2.1 and 2.3 of [22]. To observe (iv), if $\alpha \le \beta$, we have (A.6) from (5.2) and (6.1) (and the identity below of (6.1)) from [22]. In contrast, if $\alpha > \beta$, then

$$\begin{aligned} \begin{aligned} (-\varDelta )^{\gamma /2}q_{\alpha ,\beta }(t,x)&=\frac{1}{\Gamma (\alpha -\beta )}\int _{0}^{t}(t-s)^{\alpha -\beta -1}(-\varDelta )^{\gamma /2}p(s,x)ds \\&=t^{\alpha -\beta -\frac{\alpha (d+\gamma )}{2}}\frac{1}{\Gamma (\alpha -\beta )}\int _{0}^{1}(1-s)^{\alpha -\beta -1}(-\varDelta )^{\gamma /2}p(1,t^{\alpha /2}x)ds \\&=t^{\alpha -\beta -\frac{\alpha (d+\gamma )}{2}}q_{\alpha ,\beta }(1,t^{-\alpha /2}x) \end{aligned} \end{aligned}$$

by the results for the case $\alpha = \beta$ and Lamma A.5 (iii), (5.2), and (6.1) (and the identity below of (6.1)) from [22]. The lemma is proved. $\square$