Abstract
This manuscript investigates the time-fractional stochastic Keller-Segel-Navier-Stokes system in Hilbert space. This work provides a theoretical framework for analyzing cell migration by incorporating memory effects and environmental noise into the chemotactic signaling and fluid interaction. The proposed system captures key dynamics of cells respond to external gradients during directed movement. The existence of local and global mild solutions with uniqueness is studied under suitable conditions by using Banach fixed point and Banach implicit function theorem. The results are obtained in the pth moment by employing fractional calculus, stochastic analysis and Mittag-Leffler functions. Furthermore, we investigated the asymptotic stability of the proposed system as time approaches infinity.
Similar content being viewed by others
Introduction
Chemotaxis is the phenomenon that is responsible for the movement of an organism or cell in response to chemical concentration. In this paper, a mathematical system deals with the movement of oxygen-driven bacteria swimming in an incompressible fluid medium like water in response to oxygen concentration, whether positive chemotaxis (higher concentrations) or negative chemotaxis (to get away from it)1,2. The system couples the Navier-Stokes equation with the Keller-Segel equation, ultimately modeling how bacteria move towards higher concentration3 and describing the impact on the bacteria population over time. There are several studies related to chemotaxis that depend on the bacteria’s cell type. In addition, through signaling mechanisms, cells sense information about concentration gradients and migrate towards higher concentrations. Bacteria use their flagella for movement. When the flagella rotate counterclockwise, they bundle together and propel the bacterium in a straight line, and it is known as a run1,3. Another type is when the flagella rotate clockwise; they fly apart, causing the bacterium to randomly change direction. It is known as a tumble4. Due to external environments such as noises and sudden jumps, the bacterium’s flagella changes its direction during tumbling.
Fractional calculus is a significant concept of integrals and derivatives to non-integer orders5. Fractional derivatives are especially used to model anomalous diffusion and real-life applications in biology, physics, chemistry, viscoelasticity, and control theory6,7,8. Taking advantage of the Mittag-Leffler function, which is analogous to the exponential function, it plays an important role in finding the solution of fractional partial differential equations9. With the advantages of the Laplacian operator, the diffusion coefficients \(\beta (-\Delta )^{\alpha /2}\), \(\mu (-\Delta )^{\alpha /2}, \zeta (-\Delta )^{\alpha /2}\) represent the bacterial motion, how chemical consumption increases, and how bacteria are transported through the fluid u. When \(0<\alpha <2,\) the Laplacian operator represents anomalous diffusion where diffusion occurs at a non-local scale. This is used to model the long-range diffusion in chemotaxis or fluid. Moreover, we use the weak \(L^p\) space \(L^{p,\infty }\), a special case of the Lorentz space used to define interpolation between function spaces. The authors’ work in Ref. 10 investigated the existence of local and global solutions in Lebesgue space by using the fixed point theorem. The authors in Refs. 11 and 12 investigated the existence of local and global in two dimensions and three dimensions, respectively. The local and global mild solutions are obtained in \(L^p\) space in the sense of the pth moment. Moreover, the authors in Ref. 13 investigated the well-posedness of the time-fractional doubly parabolic Keller-Segel system in \(L^p\) space. In addition, global well-posedness for the proposed system is studied in critical Besov-weak-Herz space14.
Stochastic analysis becomes more useful to model the complex reactions that happen in real-life phenomena15,16. Recent studies have significant play in modelling deterministic models with stochastic effects, exhibiting inherent fluctuations in biological and physical processes. Stochastic models play a vital role when modeling the behavior of chemotactic bacteria that flow through the fluid and are influenced by stochastic perturbation. Furthermore, chemotaxis is often seen in effective host defense mechanisms in the human immune system. In response to the presence of a foreign body, the neutrophils encounter signals such as chemokines, which work as an attractant, and also neutrophils carry out their immune functions through the mechanisms from oxygen-dependent pathways to oxygen-independent17,18. During polarization, the neutrophils physically reorganize themselves in the direction of the gradient. With the significant real-life applications mentioned above, we can conclude that the evolution of velocity fluid is affected by the potential function \(n \nabla \Phi\) and random disturbances from the external environment. By considering the external forces that all affect the evolution of fluid u, we model the time-space fractional stochastic Keller-Segel-Navier-Stokes (TFSKSNS) system in (1). Initially, the author19 investigated the existence of variational and weak solutions for the two-dimensional chemotaxis Navier-Stokes equation with Gaussian noise. Continuous Gaussian noise, modeled by Brownian motion, represents microscopic-scale random perturbations, such as molecular fluctuations and small-scale force variations20. The Brownian motion in chemotaxis-fluid systems represents the effect of long-term diffusive stability, allowing for a more realistic description of bacterial dispersion under uncertain environments. In addition, stochastic chemotaxis equations with Brownian motion have employed to capture the impact of physically motivated random perturbations on cell migration. The authors in Ref. 21 investigated the existence of a global weak solution for the deterministic part of the system (1). In addition, the authors in Ref. 22 studied the existence of solutions in the sense of mild and weak solutions using a method of cutting off the stochastic system. Moreover, the authors in Ref. 23 derived a new stochastic method to find the solution of the chemotaxis-Navier-Stokes equation by using the Galerkin approximation. Furthermore, the authors investigated global weak solutions and asymptotic stabilization for the three-dimensional Keller-Segel-Navier-Stokes system with a logistic source24. In addition to continuous fluctuations, discrete stochastic events are often observed in biological systems. These are mathematically modeled using Poisson jumps, represented by a Poisson random measure that induces abrupt changes at random times. Poisson jumps are particularly relevant for describing sudden bacterial re-orientations, run-and-tumble dynamics, or abrupt changes in environmental conditions25. The jump intensity parameter controls the frequency of such events, while the jump amplitude dictates their immediate effect on system variables. Several studies modelled by Brownian motion provide a comprehensive framework for modeling both continuous micro fluctuations and discrete, high-impact events in chemotaxis-fluid systems26. Overall, stochastic chemotaxis-fluid models offer a richer description of bacterial dynamics by integrating deterministic interactions with random fluctuations10,27. This allows for a better understanding of stability, pattern formation, and transient responses in realistic biological and environmental scenarios. Because of its ubiquitous at all stages, chemotaxis study is crucial to understanding the cells’ nature. Motivated by the above works10,28, to the best of the authors’ knowledge, there is a research gap in the stability behavior and existence of local and global solutions with noises for the Time-Fractional Stochastic Keller-Segel-Navier-Stokes (TFSKSNS) system with Poisson jumps. The main aim of this paper is to study the local and global mild solution to the TFSKSNS system:
-
The TFSKSNS system is proposed with the Wiener process and Poisson jumps.
-
The Wiener process represents inherent noise in cell signaling, fluctuations in environmental factors such as oxygen concentration, or random continuous fluctuations in physiological processes such as nutrient transport.
-
The Poisson jumps occur at random, discontinuous events and irregular intervals as abrupt changes in fluid flow, external forces due to environmental disturbances, or sudden bursts in concentration. In addition, this random disturbance affects the velocity field.
-
By employing a fixed point theorem and stochastic analysis, the existence of local and global mild solutions with uniqueness is derived.
-
The main challenge is to evaluate the bilinear and nonlinear terms, such as the chemotactic gradient and fluid advection terms, while solving the global existence of the considered system.
Model descriptions & preliminaries
This section deals with basic concepts, lemmas, and an introduction of some functions. The aim of this paper is to establish the local and global mild solution of the following system (1) in \(L^p\) space. In this paper, for \(0< \eta<1, 1<\alpha \le 2\) and \(d \ge 2\), we consider the following Cauchy problem for TFSKSNS:
where \(^c_0D_t^\eta n,\ ^c_0D_t^\eta c,\ ^c_0D_t^\eta u\) are the fractional Caputo derivatives of order \(\eta\) with respect to t. Here n(x, t), c(x, t), \(u(x,t), \mathscr {P}\) denote the density of cells, oxygen concentration, velocity of the fluid, and pressure field, respectively. The positive diffusion coefficients for the corresponding density of cells, concentration, and fluid velocity are denoted by \(\beta ,\mu ,\zeta\) respectively. Here \((-\Delta )^{\alpha /2}\) represents the fractional Laplacian of order \({\alpha /2}\). Also, \(B(c)= \nabla ((-\Delta )^{\frac{-\delta }{2}}c)\) is the general potential with a singular kernel. The symbol \(\Phi\) is a time-independent potential function such as gravitational force. Here \(g:(0,\infty ) \times R^d \rightarrow L^2_0(K,H)\) where K is a real Hilbert space with the norm \(\Vert \cdot \Vert _K\) and \(L_2^0(K,H)\) denotes the space of all Q-Hilbert-Schmidt operators and \(\tilde{\pi } (dt,dz)\) denotes Poisson jumps about the Poisson process with the compensated Poisson measure \(\tilde{\pi }(dt,dz)=\pi (dt,dz)-\nu (dz)dt.\) In addition, the random noises W and \(\pi\) are defined on a complete filtered probability space \((\Omega ,\mathscr {F},\{\mathscr {F}_{t}\}_{t>0},\mathbb {P})\) with normal filtration \(\{\mathscr {F}_t\}_{t \ge 0}\). W(t) is a \(\mathscr {F}_t-\)adapted cylindrical Wiener process. \(\tilde{\eta }\) is a time-homogeneous Poisson random measure that is independent of W, defined on \([0, \infty ) \times Z\) with \(\sigma -\)finite measure \(\nu (dz)\) on a measurable space (Z, B(Z)). Here, \(n_0,c_0,u_0\) are the initial conditions, which are \(\mathscr {F}_0\)-measurable H-valued stochastic processes. The above system (1) describes the TFSKSNS system, how the cell density changes with respect to concentration, how chemo-attractants change in order to cell density, and transport through the velocity field and also describes how cell density and concentration influence the motion of the velocity field. If we replace \(^c_0D_t^\eta\) by \(\alpha =\delta\), then Eq. (1) is reduced to the parabolic-elliptic Keller-Segel system coupled with the Navier-Stokes fluid20. Let \(1<p<\infty\), \(L^p(\Omega ,R^d)\) denotes the \(L^p\) space with respect to the Lebesgue measure that consists of \(R^d\) random variables endowed with the inner product \(E(\cdot ,\cdot )\) and norm \(E\Vert \cdot \Vert\) such that
Let C(J; H) be a class of continuous functions from \(J=[0,T]\) to the Hilbert space H. Let \(C_b(J;H)\) be the class of bounded and continuous functions from J to H. The time-weighted space-time Banach space
Let \(L^{p, \kappa }(\Omega ,R^d)\), where \(1<p<\infty\) and \(1 \le \kappa \le \infty\), be Lorentz space consists of \(R^d-\)valued random variables with (quasi) norm \(\Vert \cdot \Vert _{p, \kappa }\) given by
Although the above \(\Vert \chi \Vert _{p,\kappa }\) does not satisfy the triangle inequality, there exists a norm that is equivalent to \(\Vert \chi \Vert _{p, \kappa }\). Here, \(L^{p,p}(\Omega , R^d)=L^p(\Omega ,R^d)\). In particular, in the case when \(\kappa =\infty\), the weak \(L^p\) space \(L^{p,\infty }(\Omega ,R^d)\) is also a Banach space equipped with the norm
where |D| denotes the Lebesgue measure of \(D \subset R^d\). For \(1 \le p<\infty\), we see that \(E\Vert \chi \Vert ^p_{p,\infty } \le E\Vert \chi \Vert ^p_p\). In addition, for \(1 \le \kappa <\infty\), the space \(C_0^{\infty }({R^d})\) is dense in \(L^{p, \kappa }(\Omega ,R^d)\). It is not true for \(L^{p,\infty }(\Omega ,R^d)\). The dual closure of \(C_0^{\infty }({R^d})\) in \(L^{p, \infty }(\Omega ,R^d)\) which is coincides with the space \(L^{\frac{p}{p-1},1}(\Omega ,R^d)\). The set of solenoidal vectors in \(L^{p,\infty }(\Omega ,R^d)\) is denoted by \(L^{p,\infty }_\sigma (\Omega ,R^d)=PL^{p,\infty } (\Omega ,R^d)\). Here \(P:L^{p,\infty }(\Omega ,R^d) \rightarrow L^{p,\infty }_\sigma (\Omega ,R^d)\) be a surjective map and bounded projector. The space \(C_{0,\sigma }^{\infty }({R^d})\) consists of set of all the smooth solenoidal vector fields with compact support which is dense in \(L^{p, \kappa }_\sigma (\Omega ,R^d)\). Let \(L^{\frac{p}{p-1},1}_\sigma (\Omega ,R^d)\) be the dual closure of \(C_{0,\sigma }^{\infty }({R^d})\) in \(L^{p,\infty }(\Omega ,R^d)\). Let the operator \(Q \in L(K,K)\) be defined as \(Qe_\varrho =\lambda _\varrho e_\varrho\) with tr\(Q < \infty\) where \(e_\varrho ,\ \varrho =1,2,\dots\) are eigenfunctions and \(\lambda _\varrho\) are corresponding eigenvalues. The Laplacian operator \((-\Delta )^{\alpha /2}\) of order \({\alpha /2}\) which is realized through the Fourier multiplier29:
Here B(c) is expressed with convolution of a singular kernel which is represented by
Now, we introduce the generalized definition of the Caputo fractional derivative.
Definition 1.1
Reference 30 Let X be a Banach space. For any function \(v \in L^1_{loc}((0,T);X)\), then there exists \(v_0 \in X\) with
for \(v_T \in X\).
For \(\eta >-1\), we have the following distribution \(\{g_\eta \}\) as the convolution kernels:
Here \(\Gamma (w) = \int _{0}^{\infty }e^{-t}t^{z-1}dt,\vartheta (t)\) is the standard Heaviside step function, \(\Gamma (w)\) is a Gamma function, and D represents the distributional derivative. When \(\eta \le -1\), \(g_{\eta }\) can be defined in Ref. 30. Also,
Following that, we introduce a time-reflected group:
Also, supp \(\tilde{g} \subset (\infty ,0]\) and the following equality is true for \(\delta \in (0,1)\):
Definition 1.2
Reference 30 Let \(0< \eta <1\). For \(v \in L^1_{loc}([0,T);R)\) with v has a right limit \(v(0+)\) at \(t=0\) in the sense of Definition 1.1. The \(\eta\)-th order Caputo fractional derivative of v, a distribution in \(\mathscr {D}'(R)\) with support in [0, T), is defined by
Here the fractional integral operator \(I_\alpha\) is denoted by
In addition, the \(\eta\)th-order Caputo fractional derivative of v, a distribution in \(\mathscr {D}'(R)\) with support in \((-\infty ,T]\), which is represented as
Now, we define the mappings of Caputo fractional derivatives into Banach spaces.
Definition 1.3
Reference 30 Let X be a Banach space and \(v \in L^1_{loc}((0,T);B)\). Let \(v_0 \in X\). Define the weak Caputo fractional derivative of v associated with initial data \(v_0\) to be \(^c_0D_t^\eta v \in \mathscr {D}'\) such that for any test function \(\phi \in C_c^\infty ((-\infty ,T);R)\),
where \(\mathscr {D}'=\{z|z: C^\infty _c((-\infty ,T);R) \rightarrow X \text { is a bounded linear operator}\}\). It is called the weak Caputo fractional derivative \(^c_0D_t^\eta v\) with an initial value \(v_0\) the Caputo fractional derivative of v if \(v(0+)=v_0\) in the sense of Definition 1.1 equipped with the norm of the underlying Banach space X.
Lemma 1.4
Reference 31 (Generalized \(H\ddot{o}lder's\) inequality) Assume that \(1<p_1,p_2<\infty\) such that \(\frac{1}{p_1}+\frac{1}{p_2} <1,\) \(f \in L^{p_1, \kappa _1}(\Omega ,R^d)\) and \(g \in L^{p_2, \kappa _2}(\Omega ,R^d)\). Then \(fg \in L^{p, \kappa }(\Omega ,R^d)\), where \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\) and \(\kappa \ge 1\) is any number such that \(\frac{1}{ \kappa _1}+\frac{1}{ \kappa _2} \ge \frac{1}{ \kappa }\). Furthermore,
where \(p'\) is the conjugate index of p.
Lemma 1.5
Reference 32(\(H\ddot{o}lder's\) inequality for the Lorentz spaces) Let \(1<\vartheta <\infty , 1\le \rho \le \infty\) and \(J \subset R.\) Suppose that \(1<\vartheta _0,\vartheta _1<\infty\) and \(1\le \rho _0,\rho _1 \le \infty\) satisfy \(\frac{1}{\vartheta }=\frac{1}{\vartheta _0}+\frac{1}{\vartheta _1}\) and \(\frac{1}{\rho }=\frac{1}{\rho _0}+\frac{1}{\rho _1}\) respectively. Then for every \(f \in L^{\vartheta _0,\rho _0}(J,R)\) and \(g \in L^{\vartheta _1,\rho _1}(J;R)\), it holds that \(fg \in L^{\vartheta ,\rho }(J;R)\) with the \(H\ddot{o}lder's\) inequality:
In particular, for all \(f \in L^{\vartheta ,\rho }(J;R)\) and \(g \in L^{\vartheta _1,\rho }(J;R)\) such that
with some constant \(C=C(\vartheta _0,\vartheta _1,\rho )>0\) independent of f, g and J.
Lemma 1.6
Reference 31 and 33 (Generalized Young’s inequality)
-
(a)
Assume that \(1<p_1,p_2<\infty , f\in L^{p_1, \kappa _1}(\Omega ,R^d)\) and \(g \in L^{p_2, \kappa _2}(\Omega ,R^d)\) where \(\frac{1}{p_1}+\frac{1}{p_2} >1\). Then it is true that \(f *g \in L^{p, \kappa }(\Omega ,R^d)\), where \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}-1\) and \(\kappa \ge 1\) is any number such that \(\frac{1}{ \kappa _1}+\frac{1}{ \kappa _2} \ge \frac{1}{ \kappa }\). Furthermore,
$$\begin{aligned} \Vert f*g\Vert _{p, \kappa } \le 3p \Vert f\Vert _{p_1, \kappa _1}\Vert g\Vert _{p_2, \kappa _2}. \end{aligned}$$ -
(b)
Assume that \(1<p_1<\infty , 1 \le \kappa _1 \le \kappa \le \infty ,\ f\in L^{p_1, \kappa _1}(\Omega ,R^d)\) and \(g \in L^1(\Omega ,R^d)\). Then we have
$$\begin{aligned} \Vert f*g\Vert _{p, \kappa } \le C( \kappa _1, \kappa ) \Vert f\Vert _{p_1, \kappa _1}\Vert g\Vert _1\nonumber . \end{aligned}$$ -
(c)
Assume that \(1<p_1,p_2<\infty , 1 \le \kappa _1, \kappa _2 \le \infty , \ f\in L^{p_1,\kappa _1}(\Omega ,R^d)\) and \(g \in L^{p_2,\kappa _2}(\Omega ,R^d)\), where \(p_1,p_2,\kappa _1,\kappa _2\) satisfy \(\frac{1}{p_1}+\frac{1}{ \kappa _1}=1\) and \(\frac{1}{\kappa _1}+\frac{1}{ \kappa _2}\ge 1.\) Then we get that \(f *g \in L^\infty (\Omega ,R^d)\) and
$$\begin{aligned} \Vert f*g\Vert _{\infty } \le C( \kappa _1, \kappa ) \Vert f\Vert _{p_1, \kappa _1}\Vert g\Vert _{p_2, \kappa _2}. \nonumber \end{aligned}$$
The authors in Refs. 34 and 35 stated that the fundamental solutions N(x, t) and M(x, t) of the linear non-homogeneous fractional diffusion equation can be given by taking the Fourier transform with respect to the spatial variable x and the Laplace transform with respect to time. Define N(x, t) and M(x, t) by
Lemma 1.7
Reference 34 Suppose \(0<\eta <1\) and \(1<\alpha \le 2\). Then
-
(i)
For every \(1\le p \le k_1\), and we have \(N(t,\cdot ) \in L^{p, \infty }(\Omega ,R^d)\), then there exists \(C>0\) such that
$$\begin{aligned} \Vert N(t,\cdot )\Vert _{p,\infty } \le C t^{-\frac{d\eta }{\alpha }(1-\frac{1}{p})},\ t > 0, \end{aligned}$$\(\text {where} \ k_1= \left\{ \begin{array}{ll} \frac{d}{d-\alpha } & , \text {if} \ d > \alpha , \\ \infty & , \text {otherwise}. \end{array} \right.\)
-
(ii)
For every \(1\le p \le k_2\), and we have \(M(t,\cdot ) \in L^{p,\infty }(\Omega ,R^d)\), then there exists \(C>0\) such that
$$\begin{aligned} \Vert M(t,\cdot )\Vert _{p,\infty } \le C t^{-\frac{d\eta }{\alpha }(1-\frac{1}{p}) + \eta - 1},\ t > 0, \end{aligned}$$\(\text {where} \ k_2 = \left\{ \begin{array}{ll} \frac{d}{d-2\alpha } & , \text {if} \ d > 2\alpha , \\ \infty & , \text {otherwise}. \end{array} \right.\)
-
(iii)
For every \(1\le p \le k_3\), if we have \(\nabla N(t,\cdot ) \in L^{p,\infty }(\Omega ,R^d)\), then there exists \(C>0\) such that
$$\begin{aligned} \Vert \nabla N(t,\cdot )\Vert _{p,\infty } \le C t^{-\frac{d\eta }{\alpha }(1-\frac{1}{p})-\frac{\eta }{\alpha }},\ t >0, \end{aligned}$$\(\text {where} \ k_3= \left\{ \begin{array}{ll} \frac{d}{d-\alpha +1} & , \text {if} \ d>\alpha -1, \\ \infty & , otherwise. \end{array} \right.\)
-
(iv)
For every \(1\le p \le k_4\), and we have \(\nabla M(t,\cdot ) \in L^{p,\infty }(\Omega ,R^d)\), then there exists \(C>0\) such that
$$\begin{aligned} \Vert \nabla M(t,\cdot )\Vert _{p,\infty } \le C t^{-\frac{d\eta }{\alpha }(1-\frac{1}{p})-\frac{\eta }{\alpha }+\eta -1},\ t >0, \end{aligned}$$
\(\text {where} \ k_4= \left\{ \begin{array}{ll} \frac{d}{d-2\alpha +1} & , \text {if} \ d>2\alpha -1, \\ \infty & , otherwise. \end{array} \right.\) The above lemma summarizes the weak \(L^p-\)decay estimate for the fundamental solutions N(x, t) and M(x, t).
Now, we consider operators \(P^\eta _\alpha (t),Q^\eta _\alpha (t)\) defined as follows:
Lemma 1.8
Reference 28 For \(0<\eta <1\) and \(1<\alpha \le 2\) and \(1<\kappa <\infty\). Then,
-
(i)
If \(\chi \in L^\infty (\Omega ,R^d)\), the following \(L^{\infty }\) satisfy
$$\begin{aligned} \Vert P^\eta _\alpha (t)\chi \Vert _\infty&\le \Vert \chi \Vert _\infty , \ \Vert Q^\eta _\alpha (t)\chi \Vert _\infty \le \frac{1}{\Gamma (\eta )} t^{\eta -1} \Vert \chi \Vert _\infty ,\\ \Vert \nabla P^\eta _\alpha (t)\chi \Vert _\infty&\le C t^{-\frac{\eta }{\alpha }} \Vert \chi \Vert _\infty , \ \Vert \nabla Q^\eta _\alpha (t)\chi \Vert _\infty \le C t^{-\frac{\eta }{\alpha }+\eta -1} \Vert \chi \Vert _\infty . \end{aligned}$$ -
(ii)
If \(\chi \in L^{\kappa ,\infty }(\Omega ,R^d)\) and for every \(\ell \in [\kappa , \frac{\kappa d}{d-\alpha \kappa }]\) and, if \(d>\alpha \kappa\) and \(\ell \in [\kappa ,\infty )\); else we have,
$$\begin{aligned} \Vert P^\eta _\alpha (t)\chi \Vert _{\ell ,\infty } \le Ct^{-\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })} \Vert \chi \Vert _{\kappa ,\infty }. \end{aligned}$$If \(\ell =\kappa ,\) then the constant C can be chosen to 1.
-
(iii)
If \(\chi \in L^{\kappa ,\infty }(\Omega ,R^d)\) and for every \(\ell \in [\kappa , \frac{\kappa d}{d-2\alpha \kappa }]\) and, if \(d>2\alpha \kappa\) and \(\ell \in [\kappa ,\infty )\); else we have,
$$\begin{aligned} \Vert Q^\eta _\alpha (t)\chi \Vert _{\ell ,\infty } \le Ct^{-\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })+\eta -1} \Vert \chi \Vert _{\kappa ,\infty }. \end{aligned}$$ -
(iv)
If \(\chi \in L^{\kappa ,\infty }(\Omega ,R^d)\) and for every \(\ell \in [\kappa , \frac{\kappa d}{d-\kappa (\alpha -1)}]\) and, if \(d>\kappa (\alpha -1)\) and \(\ell \in [\kappa ,\infty )\); else we have,
$$\begin{aligned} \Vert \nabla P^\eta _\alpha (t)\chi \Vert _{\ell ,\infty } \le Ct^{-\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })-\frac{\eta }{\alpha }} \Vert \chi \Vert _{\kappa ,\infty }. \end{aligned}$$ -
(v)
If \(\chi \in L^{\kappa ,\infty }(\Omega ,R^d)\) and for every \(\ell \in [\kappa , \frac{\kappa d}{d-(2\alpha - \kappa )}]\) and, if \(d>(2\alpha -1) \kappa\) and \(\ell \in [\kappa ,\infty )\); else we have,
$$\begin{aligned} \Vert \nabla Q^\eta _\alpha (t)\chi \Vert _{\ell ,\infty } \le Ct^{-\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })-\frac{\eta }{\alpha }+\eta -1} \Vert \chi \Vert _{\kappa ,\infty }. \end{aligned}$$
Now, the following lemma represents the operators \(P^\eta _\alpha (t),Q^\eta _\alpha (t)\) for the time continuity of the fundamental solutions with the help of Lemmas 1.7 and 1.8.
Proposition 1.9
Reference 35 Let \(\varPhi (x)=N(x,1)\) and \(\Psi (x)=M(x,1)=O(x,1).\) Then,
-
(i)
\(\varPhi \in L^{p,\infty }(\Omega ,R^d)\) when \(p \in [1,k_1]\) and \(\nabla \varPhi \in L^{p,\infty }(\Omega ,R^d)\) when \(p \in [1,k_2]\). \(\Psi \in L^{p,\infty }(\Omega ,R^d)\) when \(p \in [1,k_3]\) and \(\nabla \Psi \in L^{p,\infty }(\Omega ,R^d)\) when \(p \in [1,k_4]\).
-
(ii)
Assume
$$\begin{aligned} N(x,t)=t^{-\frac{d\eta }{\alpha }}\varPhi \left( \frac{x}{t^{\frac{\eta }{\alpha }}}\right) ,\ O(x,t)= t^{-\frac{d\eta }{\alpha }}\Psi \left( \frac{x}{t^{\frac{\eta }{\alpha }}}\right) ,\ M(x,t)= t^{-\frac{d\eta }{\alpha }+\eta -1}\Psi \left( \frac{x}{t^{\frac{\eta }{\alpha }}}\right) \end{aligned}$$and
$$\begin{aligned} N(x,t) \in C((0,\infty );L^{p,\infty }(\Omega ,R^d))\ \text {for} \ [1,k_1], \ \nabla N(x,t) \in C((0,\infty );L^{p,\infty }(\Omega ,R^d)) \ \text {for} \ [1,k_2], \\ M(x,t) \in C((0,\infty );L^{p,\infty }(\Omega ,R^d))\ \text {for} \ [1,k_3], \ \nabla M(x,t) \in C((0,\infty );L^{p,\infty }(\Omega ,R^d)) \ \text {for} \ [1,k_4]. \end{aligned}$$ -
(iii)
For any \(\chi \in L^{\kappa ,\infty }(\Omega ,R^d)\) with \(\ell ,\kappa\) satisfying Lemma 1.8, we have
$$\begin{aligned}&t^{\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })} P^\eta _\alpha (t)\chi \in C_b((0,\infty );L^{\ell ,\infty }(\Omega ,R^d) ),\ t^{\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })+\frac{\eta }{\alpha }} \nabla P^\eta _\alpha (t)\chi \in C_b((0,\infty );L^{\ell ,\infty }(\Omega ,R^d) ),\\&t^{\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })-\eta +1} Q^\eta _\alpha (t)\chi \in C_b((0,\infty );L^{\ell ,\infty }(\Omega ,R^d) ) ,\ t^{\frac{d\eta }{\alpha }(\frac{1}{\kappa }-\frac{1}{\ell })+\frac{\eta }{\alpha }-\eta +1} \nabla Q^\eta _\alpha (t)\chi \in C_b((0,\infty );L^{\ell ,\infty }(\Omega ,R^d) ). \end{aligned}$$In particular, when \(\ell =\kappa ,P^\eta _\alpha (t)\chi \rightarrow \chi\) in \(L^{\kappa ,\infty }(\Omega ,R^d)\) as \(t \rightarrow 0^+\).
Lemma 1.10
Reference 36 Let H be a Banach space and p(s, t) be a \(H-\)valued continuous and bounded function defined for \(0<s<t<T \le \infty .\) Let \(\zeta<1, \iota <1\) and set
Then \(g \in C_b((0,T);H)\). In addition, if \(\lim p(s,t)=\bar{p}\) exists as \((s,t) \rightarrow (0,0)\), then \(g \in C_b((0,T);H)\) with \(g(0)=\bar{p}\mathbb {B}(\zeta ,\iota )\). where \(\mathbb {B}\) is the Beta function.
Lemma 1.11
Reference 29 Let \(0<a<d, 1 \le p< \kappa <\infty , \frac{1}{\kappa }=\frac{1}{p}-\frac{1}{d}\). Then
Lemma 1.12
Reference 32 Let \(0<\tau <1\) and \(0<T \le \infty\). Then for every \(\chi \in L^{\infty }_{loc}(0,T;H)\) satisfying \(t^{\tau }\chi \in L^{\infty }((0,T);H)\), it holds that \(\chi \in L^{\frac{1}{\tau }\infty }((0,T);H)\) with the inequality
Lemma 1.13
Reference 32 Let \(0<\vartheta<\infty , 1 \le \rho \le \infty , 0<\lambda <1-\frac{1}{\vartheta }\), and \(0<T \le \infty\). Suppose that \(\chi \in L^{\vartheta _0, \rho }((0,T);H)\) and \(\psi \in L^{\vartheta _1, \rho }((0,T);Y)\)where \(1<\vartheta _0,\vartheta _1<\infty\) satisfy \(\frac{1}{\vartheta }+\lambda =\frac{1}{\vartheta _0}+\frac{1}{\vartheta _1}\). Then, the function \(\mathscr {J}_{\lambda } (\chi ,\psi )\) is defined by
it holds that \(\mathscr {J}_{\lambda }(\chi ,\psi ) \in L^{\vartheta , \rho }((0,T);R)\) with the inequality
where \(C=C(\vartheta _0,\vartheta _1,\rho ,\lambda )>0\) is a constant independent of \(T, \chi\) and \(\psi\).
Lemma 1.14
References 16,37 and 38 For any \(p \ge 1\) and any predictable \(L_2^0(K,H)\)-valued process \(\sigma (\cdot )\), then the following inequality holds:
where \(C_p\) is a constant that depends on p and T.
Lemma 1.15
References 39 and 40(Kunitha’s first inequality) For \(p \ge 2,\) there exists \(\bar{C_p} > 0\) such that
- \(H_g:\):
-
For each \(t \in (0,\infty ),\ g:(0,\infty ) \times R^d \rightarrow L^0_2(K,H)\) there exists a positive constant \(M_g\) such that
$$\begin{aligned} E\Vert g(u,t)\Vert ^p_{L^{\frac{d}{\alpha -1},\infty }_\sigma } \le M_g E\Vert u\Vert ^p. \end{aligned}$$ - \(H_{\eta }:\):
-
For \(\tilde{\eta }:(0,\infty ) \times R^d \times Z\setminus \{0\} \rightarrow R^d\) there exists \(M_{\eta }>0\) such that
$$\begin{aligned} \int _{Z} \Vert \tilde{\eta }(t,u,z)\Vert ^p_{L^{\frac{d}{\alpha -1},\infty }_\sigma } \le M_{\eta }E\Vert u\Vert ^p. \end{aligned}$$
Without loss of generality, we will consider \(\beta =\gamma =\zeta =1\) throughout the paper.
Global existence of mild solutions
This section deals with the existence of a global solution with uniqueness to the TFSKSNS system (1).
Definition 2.1
Suppose that \((n_0,c_0,u_0,\Phi )\) satisfies
An \(\{\mathscr {F}_t\}-\)adapted stochastic process (n, c, u) is considered to be a global mild solution of the system (1) on \([0,\infty )\) if
that is, \(\forall ,\ (\varsigma ,\chi ,\varPsi ) \in L^{\frac{d}{d-\alpha -\delta +2},\infty }(\Omega ,R^d) \times L^{\frac{d}{d-\alpha +1},\infty }(\Omega ,R^d) \times L^{\frac{d}{d-\alpha +1},\infty }_\sigma (\Omega ,R^d),\)
We define Banach spaces \(H_1,H_2,H_3\)
endowed with the norm
Also, we set \(H=H_1 \times H_2 \times H_3\) equipped with the norm \(E\Vert (n,c,u)\Vert ^p_H = E\Vert n\Vert ^p_{H_1} + E\Vert c\Vert ^p_{H_2}+E\Vert u\Vert ^p_{H_3}\).
Lemma 2.2
Reference 41 Let \(H_1,H_2,H_3\) be Hilbert spaces equipped with the norm \(\Vert \cdot \Vert _{H_1},\Vert \cdot \Vert _{H_2},\Vert \cdot \Vert _{H_3}\) respectively. Consider the Hilbert space \(H=H_1\times H_2 \times H_3\) with the norm
Suppose that
are continuous bilinear functions and \(L(v_1):H_1 \rightarrow H_3\) is a continuous linear map and, \(G(v_3)\subseteq H_3,\tilde{\eta }(v_3)\subseteq H_3\) are nonlinear maps. Then there exist \(C_{1,2},C_{1,3},C_{2,1},C_{2,3},C_{3,3},C_{4},C_{5},C_{6}\) such that
Define constants
Define \(B_{\varkappa }=\{(v_1,v_2,v_3) \in H: E\Vert (v_1,v_2,v_3)\Vert ^p_H \le 2K_1 \varkappa \} \ \text {where} \ 0< \varkappa < \frac{1}{4K_1K_2}\). If \(E\Vert \omega \Vert ^p_H \le \varkappa ,\) then there exists a unique solution \(v \in B_{\varkappa }\) for the equation \(v= \omega +\phi (v),\) where \(\phi (v)=(\phi _1+\phi _2+\phi _3+\phi _4+\phi _5+L+G+\tilde{\eta })(v)\).
Theorem 2.3
For \(0< \eta<1, 1<\alpha \le 2\), \(d \ge 2, 1<\delta \le d\) if \(d=2\) and \(1<\delta \le \alpha +1\) if \(d \ge 3,\) the potential function \(\Phi\) satisfies \(\nabla \Phi \in L^{\frac{d}{\alpha +1-\delta } }(\Omega ,R^d),\) and the exponents \(p,\kappa ,\ell\) satisfy either one of the following conditions: If
-
(1)
\(d=2, 1< \delta \le 2, \frac{3}{2}< \alpha \le 2, 2< 2\alpha + \delta -3\) and
$$\begin{aligned} \frac{d}{d-\alpha -1+\delta }<\kappa< \frac{d}{\delta -1},\ \frac{d}{d-\alpha -\delta +2}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa }, \kappa \le \ell < \infty . \end{aligned}$$ -
(2)
\(d=3, 2< \delta \le \alpha +1, 1< \alpha \le 2\) and
$$\begin{aligned} \frac{d}{d-\alpha +1}<\kappa<\frac{d}{\delta -1},\ \frac{d}{d-\alpha -\delta +2}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty . \end{aligned}$$ -
(3)
\(d=2, 1< \delta \le 2, \frac{3}{2}< \alpha \le 2, 2\alpha + \delta -3 \le 2\) and
$$\begin{aligned} \frac{d}{d-\alpha -1+\delta }<\kappa<\frac{d}{\delta -1},\ \frac{d}{\alpha -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty . \end{aligned}$$ -
(4)
\(d=2, 1< \delta \le 2, 1< \alpha \le \frac{3}{2}\) or \(d=3, 1< \delta \le 2,\ 1< \alpha \le 2\) or \(d \ge 4, 1< \delta \le \alpha +1, 1< \alpha \le 2\) and
$$\begin{aligned} \frac{d}{d+\delta -2} \le \kappa<\frac{d}{\delta -1},\ \frac{d}{\alpha -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty , \end{aligned}$$then for any \(n_0 \in L^{\frac{d}{\alpha +\delta -2},\infty }(\Omega ,R^d),\ c_0 \in L^{\frac{d}{\alpha -1},\infty }(\Omega ,R^d),\ u_0 \in L^{\frac{d}{\alpha -1},\infty }_\sigma (\Omega ,R^d)\), then there exists a unique mild solution (n, c, u) on H in the sense of Definition 2.1 for the TFSKSNS system (1).
Proof
We represent the mild solution (n, c, u) as the following form:
where
Next, we show that \(\begin{aligned}\phi_1(n,c) &: H_1 \times H_2 \longrightarrow H_1, \\\phi_2(u,n) &: H_3 \times H_1 \longrightarrow H_1, \\\phi_3(n,c) &: H_1 \times H_2 \longrightarrow H_2, \\\phi_4(u,n) &: H_3 \times H_1 \longrightarrow H_1, \\\phi_5(u,u) &: H_3 \times H_3 \longrightarrow H_3 .\end{aligned}\) are continuous bilinear functions, \(L(n) : H_1 \longrightarrow H_3\) is a continuous linear map and \(G(u)\subseteq H_3, \ \tilde{\eta }(u)\subseteq H_3\) are nonlinear maps. By using Lemma 1.4, for \(\frac{1}{h_1}=\frac{\alpha -1}{d}+\frac{1}{\ell }\) and \(\frac{1}{h_3}=\frac{1}{\kappa }+\frac{1}{\ell }\) , we obtain
Using Lemma 1.11, the term B(c) is given by
By using Lemma 1.8 and the above inequalities (10) and (11) we estimate the term \(\phi _1(n,c)\)
and
Set
In view of Proposition 1.9 and Eq. (12) provided that \(\psi _1(s,t) \in C_b((0,\infty );L^{\frac{d}{\alpha +\delta -2},\infty }(\Omega ,R^d)).\) By using Lemma 1.10 hence,
Also, \(t^{\frac{d\eta }{\alpha }\left( \frac{\alpha +\delta -2}{d} -\frac{1}{\kappa }\right) }E\Vert \phi _1(n, c)\Vert ^p \in C_b((0,\infty );L^{\kappa ,\infty }(\Omega ,R^d)).\) Similarly, for \(\frac{1}{h_2}=\frac{\alpha +\delta -2}{d}+\frac{1}{p}\) and \(\frac{1}{h_4}=\frac{1}{\kappa }+\frac{1}{p}\) and by using Lemma 1.4 , we obtain that
In similar way of choosing,
Now, we prove that
Now,
and
Therefore,
Again by using Lemma 1.8 and Proposition 1.9, we obtain
and
From Eqs. (12)–(16), we get that
Similarly, by using Lemma 1.4, for \(\frac{1}{h_5}=\frac{\alpha +\delta -2}{d}+\frac{1}{\ell }\) and \(\frac{1}{h_7}=\frac{1}{\kappa }+\frac{1}{\ell }\), we obtain that
Again choose,
Now,
and
Hence, we obtain
Similarly, for \(\frac{1}{h_6}=\frac{\alpha -1}{d}+\frac{1}{p}\) and \(\frac{1}{h_8}=\frac{1}{\ell }+\frac{1}{p}\) and by using Lemmas 1.4 and 1.8, we obtain that
Again by choosing,
Now, we prove that
and
Therefore,
Again by using Lemma 1.8 and Proposition 1.9, we obtain,
and
From Eqs. (18)–(23), we get that
Let \(1<k<\infty\), since P is a bounded projector from \(L^{k,\infty }\) onto \(L_\sigma ^{k,\infty }\), then by using Lemmas 1.4 and 1.8, for \(\frac{1}{h_9}=\frac{\alpha -1}{d}+\frac{1}{p}\) and \(\frac{1}{h_{11}}=\frac{1}{p}+\frac{1}{p}\) , we obtain
and
Similarly, for \(\frac{1}{h_{10}}=\frac{1}{\kappa }+\frac{\alpha +1-\delta }{d}\) and \(\frac{1}{h_{12}}=\frac{1}{\kappa }+\frac{\alpha +1-\delta }{d}\) and by using Lemma 1.4, we obtain that
and
Once again by using Lemmas 1.4 and 1.8, and asummption \(H_g\), for \(\frac{1}{h_{13}}=\frac{\alpha }{d}+\frac{1}{p}\) and \(\frac{1}{h_{15}}=\frac{1}{d}+\frac{2}{p}\), we obtain
where \(C_{5}=C^p M_g \mathbb {B} \left( -\frac{d\eta p}{\alpha }\left( \frac{1}{p}-\frac{\alpha -1}{d}\right) -p+1,1 -\frac{d\eta }{\alpha }\left( \frac{\alpha -1}{d}-\frac{1}{p} \right) \right)\).
Using the assumption \(H_g\), we obtain
where \(\tilde{C}_{5}=C^p M_g \mathbb {B} \left( -\frac{d\eta p}{\alpha }\left( \frac{1}{p}-\frac{\alpha -1}{d}\right) -p+1,1 -\frac{d\eta }{\alpha }\left( \frac{\alpha -1}{d}-\frac{1}{p} \right) \right)\).
Once again by using Lemmas 1.4 and 1.8, and assumption \(H_{\eta }\), for \(\frac{1}{h_{14}}=\frac{1}{d}+\frac{1}{p}\) and \(\frac{1}{h_{16}}=\frac{1}{d}+\frac{2}{p}\) , we obtain
and by using Lemma 1.15 and assumption \(H_{\eta }\),
Again, by choosing suitable functions,
By utilizing Proposition 1.9 and Lemma 1.10, we get
Again by using Proposition 1.9 and Lemma 1.10 we obtain,
and
From Eqs. (25)–(33), we get that
Define ball \(B_{k}=\{(n,c,u) \in H:E\Vert (n,c,u)\Vert ^p_H \le 2K_1 \varkappa \}\) in H and
then we get
By using Lemma 2.2, there exists a unique global mild solution \((n,c,u) \in B_\varkappa\) of TFSKSNS Eq. (1) with \(0<\varkappa < \frac{1}{4K_1K_2}\). Suppose \(\varsigma ,\chi ,\varPsi \in C^\infty _0(R^d)\) and by estimate of Eq. (12), we obtain
By dominated convergence theorem, we get
In the same way of estimates (13),(15),(20),(22),(26),(28),(30) and (32) we get
Thus (n(t), c(t), u(t)) converges weakly to \((n_0,c_0,u_0)\) as \(t \rightarrow 0^+.\) Therefore, we proved that
This completes the proof. \(\square\)
Corollary 2.4
(Lorentz regularity in time direction) Let \(\frac{1}{\varepsilon _1}=\frac{d\eta }{\alpha }(\frac{\alpha +\delta -2}{d} -\frac{1}{\kappa }), \ \frac{1}{\varepsilon _2}=\frac{d\eta }{\alpha }(\frac{\alpha -1}{d} -\frac{1}{\ell }), \ \frac{1}{\varepsilon _3}=\frac{d\eta }{\alpha }(\frac{\alpha -1}{d} -\frac{1}{p}).\) Then the mild solution (n, c, u) of (1) obtained in Theorem 2.3 also satisfies Lorentz regulariy in time direction
Proof
Let \(\frac{1}{\varepsilon _1}=\frac{d\eta }{\alpha }(\frac{\alpha +\delta -2}{d} -\frac{1}{\kappa }), \ \frac{1}{\varepsilon _2}=\frac{d\eta }{\alpha }(\frac{\alpha -1}{d} -\frac{1}{\ell }), \ \frac{1}{\varepsilon _3}=\frac{d\eta }{\alpha }(\frac{\alpha -1}{d} -\frac{1}{p})\). In Theorem 2.3, the inequality (13) holds that
By using Lemma 1.13, we obtain \(E\Vert \phi _1(n,c)\Vert ^p_{L^{\varepsilon _1,\infty }} \le C^p E\Vert n\Vert ^p_{L^{\varepsilon _1,\infty }(L^{\kappa ,\infty })}E\Vert c\Vert ^p_{L^{\varepsilon _2,\infty }(L^{\ell ,\infty })}.\)
Similarly, we estimate the inequality (13),(15),(20),(26),(28),(30),(30),(32) and utilizing Lemma 1.13 gives that
Moreover, by using Lemma 1.13, we obtain
By Lemma 2.2, we proceed in the same way as in Theorem 2.3, we obtain that the mild solution (n, c, u) of (1) satisfies the Lorentz regularity in time direction:
\(\square\)
Local existence of mild solution
This section deals with the local existence and uniqueness of mild solutions for the TFSKSNS system (1).
Definition 3.1
Let \((n_0,c_0,u_0,\Phi )\) satisfy
An \(\{\mathscr {F}_t\}-\)adapted stochastic process (n, c, u) is said to be a local mild solution of the system (1) on [0, T] if
as \(\ t \rightarrow 0^+,\) that is , \(\forall ,(\varsigma ,\chi ,\varPsi ) \in L^{\kappa ',1}(\Omega ,R^d) \times L^{\ell ',1}(\Omega ,R^d) \times L^{p',1}_\sigma (\Omega ,R^d),\)
Theorem 3.2
For \(0< \eta<1, 1<\alpha \le 2\), \(d \ge 2, 1<\delta \le d\) if \(d=2\) and \(1<\delta \le \alpha +1\) if \(d \ge 3,\) the potential function \(\Phi\) satisfies \(\nabla \Phi \in L^{\frac{d}{\alpha +1-\delta } }(\Omega ,R^d),\) and the exponents \(p,\kappa ,\ell\) satisfy either one of the following conditions:
-
(1)
If \(d=2, 1< \delta \le 2, \frac{3}{2} < \alpha \le 2\) and
$$\begin{aligned} \frac{d}{d-\alpha -1+\delta }<\kappa< \frac{d}{\delta -1},\ \frac{d}{\alpha -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa }, \kappa \le \ell < \infty . \end{aligned}$$ -
(2)
\(d=3, 2< \delta \le \alpha +1, 1< \alpha \le 2\) and
$$\begin{aligned} \frac{2d}{d+\delta -1}<\kappa<\frac{d}{d-\alpha +1},\ \frac{\kappa }{\kappa -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty , \end{aligned}$$or
$$\begin{aligned} \frac{d}{d-\alpha +1}<\kappa<\frac{d}{\delta -1},\ \frac{d}{\alpha -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty .\end{aligned}$$ -
(3)
\(d=2, 1< \delta \le 2, 1< \alpha \le \frac{3}{2}\) or \(d=3, 1< \delta \le 2, 1< \alpha \le 2\) or \(d \ge 4, 1< \delta \le \alpha +1, 1< \alpha \le 2\) and
$$\begin{aligned} \frac{d}{d+\delta -2}<\kappa<\frac{d}{\delta -1},\ \frac{d}{\alpha -1}<\ell ,p<\frac{\kappa d}{d-(\delta -1)\kappa },\kappa \le \ell < \infty \end{aligned},$$then for any \(n_0 \in L^{\frac{d}{\alpha +\delta -2},\infty }(\Omega ,R^d) ,\ c_0 \in L^{\frac{d}{\alpha -1},\infty }(\Omega ,R^d), \ u_0 \in L^{\frac{d}{\alpha -1},\infty }_\sigma (\Omega ,R^d)\) and \(T>0\) then there exists a unique local mild solution (n, c, u) in the sense of Definition 3.1 for the TFSKSNS system (1).
Proof
We define Banach spaces \(H_{1,T},H_{2,T},H_{3,T}\) as:
endowed with the norm
Set \(H_T=H_{1,T} \times H_{2,T} \times H_{3,T}\) equipped with the norm
The notations \(h_3,h_4,h_7,h_8,h_{11},h_{12},h_{15},h_{16}\) are taken as the same in the proof of Theorem 2.3. With the help of Lemma 1.4 and 1.8, we have,
By Lemma 1.8, we have
Let \(B_{\bar{\varkappa }}=\{(n,c,u)\in H_T: E\Vert (n,c,u)\Vert ^p_{H} \le 2 \bar{K}_1\bar{\varkappa }\}\). Combining the above inequalities (35–42), we obtain
where
For any initial data \((n_o,c_0,u_0) \in L^{\frac{d}{\alpha +\delta -2},\infty }(\Omega ,R^d) \times L^{\frac{d}{\alpha -1},\infty }(\Omega ,R^d) \times L^{\frac{d}{\alpha -1},\infty }_\sigma (\Omega ,R^d).\) Then, there exists a time \(T^{*}\) small enough such that \(\bar{K}_2\) small enough to satisfy \(4\bar{K}_1 \bar{\varkappa } < \frac{1}{\bar{K}_2}\). Then by Lemma 2.2, the TFSKSNS system (1) has a unique mild solution on \([0,T ^*]\). \(\square\)
Asymptotic stability
In this section, we prove the asymptotic stability of the mild solution to the TFSKSNS system (1).
Theorem 4.1
Suppose the assumptions of Theorem 2.3 are fulfilled. Assume (n, c, u) and \((\tilde{n},\tilde{c},\tilde{u})\) are two mild solutions obtained by Theorem 2.3 with initial data \((n_0,c_0,u_0)\) and \((\tilde{n}_0,\tilde{c}_0,\tilde{u}_0)\) such that
if and only if
Proof
Set
By Eq. (12), we see that
Similarly, we estimate the above inequality as (12) and (14),
Then Eq. (45) becomes
Again we have,
Suppose that \((n,c,u),(\tilde{n},\tilde{c},\tilde{u}) \in B_{\varkappa }\) such that \(E\Vert (n,c,u)\Vert ^P_H,E\Vert \tilde{n},\tilde{c},\tilde{u}\Vert ^p_X \le 2K_1 \varkappa .\) Adding the above inequalities (46–51), and by applying (44) we obtain
It follows that \(S_1=S_2=S_3=S_4=S_5=S_6=0\) since \(4K_1K_2 \varkappa <1.\)
We have to prove Eq. (43) implies Eq. (44). By the assumption \(S_1=S_2=S_3=S_4=S_5=S_6=0\) and we use the estimates (46-51), we get
This completes the proof. \(\square\)
Remark 4.2
Brownian motion modulates the asymptotic stability of the system by continuous, small fluctuations around the steady state. While the system remain close to its long-term state on average, these stochastic perturbations cause mild, persistent deviations, slightly reducing the rate of convergence. In other words, the system exhibits stochastic asymptotic stability, where cell migration remain bounded and fluctuate around the steady state over time due to microscopic-scale molecular noise and random forces in the chemotactic response.
Remark 4.3
Poisson jumps influence the asymptotic stability of the system in a more abrupt and discrete manner. Each jump represents a sudden, instantaneous disturbance that instantaneously perturbs the cell population away from its long-term state. Biologically, these jumps are interpreted as abrupt changes in the movement direction of individual cells or small groups of bacteria, caused by sudden environmental stimuli, collisions, and random signaling events in the chemotactic field. As a result, the collective cell migration exhibits sharp deviations from the steady-state distribution, producing transient instabilities in both cell density and orientation.
Although the system eventually returns toward the long-term equilibrium on average, the intensity of the jumps lead to pronounced, deviated from the steady state, transiently altering the preferred direction of movement of the bacterial population. In other words, while the system remains bounded and tends to the steady state in the long run, individual cells are abruptly change direction, causing sudden fluctuations in density, concentration, veloccity and directional alignment. This behavior illustrates stochastic asymptotic stability under jump noise, where the system remains stable in the long-term sense but exhibits sudden, discrete deviations in response to environmental perturbations.
Remark 4.4
In KSNS system, the stochastic components of the system have distinct physical meanings. The Brownian motion represents continuous microscopic fluctuations in the velocity field and chemical concentration, leading to smooth and persistent random variations in bacterial velocity and orientation. This corresponds to the natural irregularities observed during the bacterium’s run phase. In contrast, the Poisson jumps model sudden and discontinuous events that cause instantaneous changes in the system’s state. Biologically, such jumps represent abrupt re-orientations of bacterial flagella during the tumble phase, collisions with other particles, and rapid local and global changes in the system. These jumps effects introduce discontinuities in the velocity field, resulting in transient deviations from steady state before the system tends to stability. The inclusion of both Brownian motion and Poisson jumps in the chemotactic field captures the dual nature of bacterial motion such as the continuous random drift due to molecular fluctuations and the abrupt directional changes triggered by environmental stimuli and interactions. Thus providing a more realistic description of bacterial dynamics under stochastic influences.
Numerical results
In this section, we present numerical simulation of TFSKSNS equation with Wiener process and Poisson jumps. Consider a 2D spatial domain \([0,1] \times [0,1].\) The non local condition captured by fractional Laplacian operator of order \(\alpha =1.5\). The fractional Caputo derivatives of order \(\eta =0.8\). The diffusion coefficients of density n, oxygen concentration c and velocity u is denoted by \(\beta = 0.1, \mu = 0.1\) and \(\zeta = 0.1\). The time step \(\Delta t=0.01\) and spatial step is \(\Delta x=\Delta y =0.05.\) The initial conditions are
Moreover, the Wiener process is scaled by a factor \(g=0.05\) and Poisson process is scaled by the coefficient \(\tilde{\eta }=2\).
Bacteria depend on oxygen to survive and move in a shallow, two-dimensional fluid environment. They move in the direction of areas with greater oxygen concentrations, exhibiting chemotactic behavior. Furthermore, the flow of the surrounding fluid affects their mobility. We visualize phenomena such as aggregation of bacteria, fluid vortices caused by bacterial forcing, and the interplay of diffusion, chemotaxis, and fluid dynamics. The proposed example is divided into two cases. In the first case, the numerical simulation take place up to time step 1 and in the second case, the numerical simulation take place up to time step 3. The finite difference method is used to approximate the spatial derivatives. The evolution of density, oxygen concentration and velocity dynamically over time is influenced by stochastic noise and Poisson jumps.
Case:1
Dynamical behaviour of bacteria with respect to density n.
Figure 1 shows the density behaviour of the bacteria with the above mentioned fixed parameter.
Dynamical behaviour of bacteria with respect to oxygen concentration c.
Figure 2 shows behaviour of oxygen concentration with the fixed parameter. Also, the oxygen concentration exhibits the advection and diffusion reaction in the proposed system. We show the time-varying dynamic patterns of bacterial density and oxygen concentration in Figs. 1 and 2.
Dynamical behaviour of bacteria with respect to velocity \(u_x\).
Dynamical behaviour of bacteria with respect to velocity \(u_y\).
In addition, the dynamic variation of bacterial due to velocity is shown in Figs. 3 and 4. The velocity \(n_x(x,y,t)\) and \(n_y(x,y,t)\) is influenced by the density and oxygen concentration of the bacteria. Also, the velocity field adopt the density and concentration of oxygen. However, with respect to the time and chemical concentration, the oxygen concentration consumes due to the movement of bacteria in the fluid medium. Due to the external force, disturbances and jumps, the cells of the bacteria moves towards the bottom of the fluid. Hence, the maximum values of \(n,c,u_x\) and \(u_y\) verifies numerical stability at each time step is given by observation in
Remark 5.1
Numerical approximations for the chemotaxis-Navier-Stokes equation with optimal error estimate is provided to check the theoretical results in Ref. 42. The convergence rate of density, concentration and velocity at various step size is provided.
Case: 2
Dynamical behaviour of bacteria with respect to density n.
Dynamical behaviour of bacteria with respect to oxygen concentration c.
Similarly, the dynamical behaviour of bacteria with respect to density n, oxygen concentration c and velocity field u is presented in the Figs. 5, 6, 7 and 8, respectively. In Fig. 5, the density distribution n(x, t) illustrates the temporal evolution of bacterial concentration under chemotactic influence. Localized aggregation zones emerge, indicating collective movement of bacteria toward regions of higher oxygen concentration. Moreover, the Fig. 6 shows that the oxygen concentration c(x, t) decreases notably in areas of dense bacterial clustering, reflecting oxygen consumption during metabolic activity. This spatial pattern highlights the strong coupling between bacterial chemotaxis and oxygen diffusion within the domain. Also, Fig. 7 represents the velocity field \(u_x(x,t)\) demonstrates the horizontal flow pattern influenced by bacterial movement and stochastic fluctuations. Distinct vortical regions appear, revealing the influence of bacterial motility on the surrounding fluid environment. Further, Fig. 8 shows that the vertical component of the velocity field, \(u_y(x,t)\), depicts the upward and downward flow dynamics within the fluid. The variation in \(u_y\) emphasizes the vertical transport mechanisms generated by bacterial activity.
Dynamical behaviour of bacteria with respect to velocity \(u_x\).
Dynamical behaviour of bacteria with respect to velocity \(u_y\).
The maximum values of \(n,c,u_x\) and \(u_y\) is verifies numerical stability at each time step is given by the following observation
Remark 5.2
Numerical examples are provided for the proposed system with particular parameters only. The authors cannot provide comparison results. The exact solutions to the proposed system will be obtained and compared with other methods in future works.
Conclusion
A theoretical approach to the existence of solutions to the time-fractional stochastic Keller-Segel-Navier-Stokes equation with noises has been investigated in Hilbert space. In addition, we proved the existence of local and glocal mild solutions and uniqueness with noises by using Banach fixed point and Banach implicit function theorem. The tools from stochastic analysis, fractional calculus, and Mittag-Leffler functions were employed. Further, the asymptotic stability results for the considered system in Hilbert space are obtained. Finally, the study focuses on the numerical example of considered system with numerical results. Future studies will focus on the time-space fractional stochastic Keller-Segel-Navier-Stokes equation with impulses and a delay term. Furthermore, the proposed numerical schemes’ error analysis will be investigated to ensure accuracy.
Data availability
All data generated or analysed during this study are included in this published article.
References
Bren, A. & Eisenbach, M. How signals are heard during bacterial chemotaxis: Protein-protein interactions in sensory signal propagation. J. Bacteriol. 182, 6865–6873 (2000).
Sourjik, V. & Wingreen, N. S. Responding to chemical gradients: Bacterial chemotaxis. Curr. Opin. Cell Biol. 24, 262–268 (2012).
Adler, J. Chemotaxis in bacteria. Science 153, 708–716 (1966).
Webre, D. J. et al. Bacterial chemotaxis. Curr. Biol. 13, R47–R49 (2003).
Podlubny, I. Fractional Differential Equations (Academic Press, 1999).
Sharma, A., Mishra, S. N. & Shukla, A. Gronwall’s inequality and stability analysis of nonlinear fractional difference equations. J. Nonlinear Complex Data Sci. 26(2), 79–94 (2025).
Sharma, A., Mishra, S. N. & Shukla, A. Finite-time stability and attractiveness of uncertain nonlinear nabla fractional systems involving Hilfer-type operators. Complex Anal. Oper. Theory 19(6), 150 (2025).
Gautam, P. & Shukla, A. Stochastic controllability of semilinear fractional control differential equations. Chaos Solitons Fract. 174, 113858 (2023).
Miller, K. S. & Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, 1993).
Jiang, Z. & Wang, L. Mild solutions to the Cauchy problem for time-space fractional Keller-Segel-Navier-Stokes system. Preprint at https://arxiv.org/abs/2201.00000 (2022).
Chae, M., Kang, K. & Lee, J. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discret. Contin. Dyn. Syst. 33, 2271–2297 (2011).
Chae, M., Kang, K. & Lee, J. Global existence and temporal decay in Keller-Segel models coupled to fluid equations. J. Differ. Equ. https://doi.org/10.1080/03605302.2013.852224 (2013).
Bezerra, M. et al. On the fractional doubly parabolic Keller-Segel system modelling chemotaxis. Sci. China Math. https://doi.org/10.1007/s11425-020-1846-x (2021).
Azevedo, J. et al. Well-posedness and asymptotic behavior for the fractional Keller-Segel system in critical Besov-Herz-type spaces. Math. Methods Appl. Sci 45, 6268–6287 (2022).
Mao, X. Stochastic Differential Equations and Applications 2nd edn. (Horwood Publishing, 2007).
Da Prato, G. & Zabczyk, J. Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). https://doi.org/10.1017/CBO9781107295513.
Ball, G. F. M. Encyclopedia of Food Sciences and Nutrition (Elsevier, 2003).
Petri, B. & Sanz, M. J. Neutrophil chemotaxis. Cell Tissue Res. 371, 425–436 (2018).
Zhai, J. & Zhang, T. Stochastic chemotaxis-Navier-Stokes system. J. Math. Pures Appl. 107, 1 (2017).
Kozono, H., Miura, M. & Sugiyama, Y. Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid. J. Differ. Equ. 267, 5410–5492 (2019).
Liu, J. & Lorz, A. A coupled chemotaxis-fluid model: global existence. Ann. Inst. H. Poincaré 28, 643–652 (2011).
Zhai, J. & Zhang, T. 2D stochastic chemotaxis-Navier-Stokes system. J. Math. Pures Appl. 138, 307–355 (2020).
Hausenblas, E., Moghomye, B. J. & Razafimandimby, P. On the existence and uniqueness of solution to a stochastic chemotaxis-Navier-Stokes model. Preprint at https://arxiv.org/abs/2301.00654 (2023).
Winkler, M. A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization. J. Funct. Anal. 276, 1339–1401 (2019).
Zhang, L. & Liu, B. On the Keller-Segel models interacting with a stochastically forced incompressible viscous flow in \(\mathbb{R} ^2\). J. Differ. Equ. 414, 487–554 (2025).
Lorz, A. Coupled chemotaxis-fluid model. Math. Models Methods Appl. Sci. 20(6), 987–1004 (2010).
Wang, J., Chen, H. & Zhuang, M. Global existence of solutions to a Keller-Segel model with logistic source. Discrete Contin. Dyn. Syst. B 30(8), 2806–2821. https://doi.org/10.3934/dcdsb.2024191 (2025).
Jiang, Z. & Wang, L. Existence and asymptotic stability of mild solution to fractional Keller-Segel-Navier-Stokes system. Math. Meth. Appl. Sci. 47, 9814–9833 (2024).
Stein, E. M. Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, 1970).
Li, L. & Liu, J. A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM J. Math. Anal. https://doi.org/10.48550/arXiv.1612.05103 (2018).
O’Neil, R. Convolution operators and \(L^{(p, q)}\) spaces. Duke Math. J. 30, 129–142 (1963).
Takeuchi, T. Various regularity estimates for the Keller-Segel-Navier-Stokes system in Besov spaces. J. Differ. Equ. 343, 606–658 (2023).
Blozinski, A. P. On a convolution theorem for \(L^{(p, q)}\) spaces. Trans. Am. Math. Soc. 164, 255–265 (1972).
Kemppainen, J., Siljander, J. & Zacher, R. Representation of solutions and large-time behaviour for fully nonlocal diffusion equations. J. Differ. Equ. 263, 149–201 (2017).
Jiang, Z. & Wang, L. Well-posedness of Keller-Segel-Navier-Stokes equations with fractional diffusion in Besov spaces. Z. Angew. Math. Phys. https://doi.org/10.1007/s00033-024-02268-x (2024).
Kato, T. Strong solutions of the Navier-Stokes equation in Morrey spaces. Bol. Soc. Bras. Mat. 22, 127–155 (1992).
Prato, G. D. & Zabczyk, J. Stochastic Equations in Infinite Dimensions (Cambridge University Press, 2014).
Ansari, S., Malik, M. & Dhayal, R. Optimal controls for multi-term fractional stochastic integro-differential equations with impulses and infinite delay. Int. J. Syst. Sci. 56(8), 1862–1883 (2025).
Kunita, H. Stochastic Differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In Real and Stochastic Analysis, 305-373 (Birkhäuser, 2004).
Applebaum, D. Lévy Processes and Stochastic Calculus (Cambridge University Press, 2009).
Ferreira, L. C. F. & Postigo, M. Global well-posedness and asymptotic behaviour in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids. J. Math. Phys. 60, 061502 (2019).
Duarte-Rodríguez, A. et al. Numerical analysis for a chemotaxis-Navier-Stokes system. ESAIM Math. Model. Numer. Anal. 55, S415–S445 (2020).
Dombrowski, C. et al. Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103 (2004).
Tuval, I. et al. Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. U.S.A. 102, 2277–2282 (2005).
Funding
Open access funding provided by Vellore Institute of Technology. This work is supported by Science and Engineering Research Board (SERB), DST, Govt. of India, SPG project File No: SPG/2021/002891 dated 30.08.2022.
Author information
Authors and Affiliations
Contributions
Divyabala K: Writing—original draft; visualization; methodology; software; investigation; conceptualization. Durga N: Conceptualization; formal analysis; funding acquisition; methodology; supervision; validation; writing—review and editing.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Ethical Statement
The authors declare that they comply with ethical standards.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Divyabala, K., Durga, N. Existence and stability of time-fractional Keller-Segel-Navier-Stokes system with Poisson jumps. Sci Rep 16, 5369 (2026). https://doi.org/10.1038/s41598-025-28809-6
Received:
Accepted:
Published:
Version of record:
DOI: https://doi.org/10.1038/s41598-025-28809-6
