Abstract
On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
Not applicable.
References
Barbu, V., Da Prato, G., Röckner, M.: Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57(1), 187–211 (2008)
Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Lecture Notes in Mathematics. Springer, Cham (2016)
Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin–New York (1976)
Bianchi, L.A., Blömker, D., Yang, M.: Additive noise destroys the random attractor close to bifurcation. Nonlinearity. 29(12), 3934–3960 (2016)
Birkhoff, G., Rota, G.-C.: Ordinary Differential Equations, 4th edn. John Wiley & Sons Inc, New York (1989)
Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Brzeźniak, Z., Hausenblas, E., Razafimandimby, P.A.: Stochastic reaction diffusion equations driven by jump processes. Potential Anal. 49, 131–201 (2018)
Brzeźniak, Z.: Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4(1), 1–45 (1995)
Brzeźniak, Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61(3–4), 245–295 (1997)
Brzeźniak, Z., Motyl, E.: Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D domains. J. Differential Equations. 254(4), 1627–1685 (2013)
Brzeźniak, Z., Hausenblas, E., Motyl, E.: Uniqueness in law of the stochastic convolution process driven by Lévy noise. Electron. J. Probab. 18, 57–15 (2013)
Cao, Y., Erban, R.: Stochastic Turing patterns: analysis of compartment based approaches. Bull. Math. Biol. 76(12), 3051–3069 (2014)
Capiński, M.: A note on uniqueness of stochastic Navier-Stokes equations. Univ. Iagel. Acta Math. XXX, 219–228 (1993)
Capiński, M., Peszat, S.: Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations. NoDEA Nonlinear Differential Equations Appl. 4(2), 185–200 (1997)
Crauel, H., Flandoli, F.: Additive noise destroys a pitchfork bifurcation. J. Dynam. Differential Equations. 10(2), 259–274 (1998)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Encyclopedia of mathematics and its applications, vol. 152. Cambridge University Press, Cambridge (2014)
Dareiotis, K., Gerencsér, M., Gess, B.: Porous media equations with multiplicative space-time white noise. Ann. Inst. Henri Poincaré (B) Probab. Stat. 57(4), 2354–2371 (2021)
Debussche, A., Högele, M., Imkeller, P.: The Dynamics of Nonlinear Reaction-diffusion Equations with Small Lévy noise. Lecture Notes in Mathematics, vol. 2085. Springer, Cham (2013)
Dhariwal, G., Huber, F., Jüngel, A., Kuehn, C., Neamţu, A.: Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method. Ann. Inst. Henri Poincaré (B) Probab. Stat. 51(1), 577–602 (2021)
Dillon, R., Maini, P.K., Othmer, H.G.: Pattern formation in generalized Turing systems. I. Steady-state patterns in systems with mixed boundary conditions. J. Math. Biol. 32(4), 345–393 (1994)
Duan, J., Wang, W.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier Insights. Elsevier, Amsterdam (2014)
Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002). Revised reprint of the 1989 original
Dulos, E., Boissonade, J., Perraud, J.J., Rudovics, B., De Kepper, P.: Chemical morphogenesis: turing patterns in an experimental chemical system. Acta Biotheor. 44(3–4), 249–261 (1996)
Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1986)
Evans, L.C.: An Introduction to Stochastic Differential Equations. American Mathematical Society, Providence (2012)
Fehrman, B., Gess, B.: Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise. Journal de Mathématiques Pures et Appliquées. 148, 221–266 (2021)
Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models. Lecture Notes in Mathematics, vol. 2015. Springer, Heidelberg (2011)
Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields. 102(3), 367–391 (1995)
Gess, B.: Optimal regularity for the porous medium equation. J. Eur. Math. Soc. 23(2), 425–465 (2021)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chemical Engineering Science. 38(1), 29–43 (1983)
Grieser, D.: Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Comm. Partial Differential Equations. 27(7–8), 1283–1299 (2002)
Hastings, A., Abbott, K.C., Cuddington, K., Francis, T., Lai, Y.-C., Morozov, A., Petrovskii, S., Zeeman, M.L.: Effects of stochasticity on the length and behaviour of ecological transients. R. Soc. Interface 18(20210257), 1–12 (2021)
Hausenblas, E., Panda, A.A.: Correction to: The stochastic Gierer Meinhardt system. Appl. Math. Optim. 86(2), 20–1 (2022)
Hausenblas, E., Panda, A.A.: The stochastic Gierer-Meinhardt system. Appl. Math. Optim. 85(2), 11–49 (2022)
Hausenblas, E., Randrianasolo, T.A., Thalhammer, M.: Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations. J. Comput. Appl. Math. 364, 112335–27 (2020)
Hörmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Classics in Mathematics. Springer, Berlin (2007). Pseudo-differential operators, Reprint of the 1994 edition
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Probability and its Applications. Springer, New York (2002)
Kealy, B.J., Wollkind, D.J.: A nonlinear stability analysis of vegetative turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat environment. Bulletin of Mathematical Biology. 74(4), 803–833 (2012)
Klausmeier, C.A.: Regular and irregular patterns in semi-arid vegetation. Science. 284, 1826–1828 (1999)
Kotelenez, P.: Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Related Fields. 93(1), 1–19 (1992)
Kotelenez, P.: Stochastic Ordinary and Stochastic Partial Differential Equations. Stochastic Modelling and Applied Probability, vol. 58. Springer, New York (2008). Transition from microscopic to macroscopic equations
Kurtz, T.G.: The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12, 951–965 (2007)
Levin, S.A., Segel, L.A.: Hypothesis for origin of planktonic patchiness. Nature 259, 659 (1976)
Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88(3), 309–318 (1983)
Liu, W.: Harnack inequality and applications for stochastic evolution equations with monotone drifts. Journal of Evolution Equations. 9(4), 747–770 (2009)
Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Universitext. Springer, Cham (2015)
Ma, T., Zhu, R.: Wong-Zakai approximation and support theorem for SPDEs with locally monotone coefficients. J. Math. Anal. Appl. 469(2), 623–660 (2019)
Maini, P.K., Woolley, T.E.: The Turing model for biological pattern formation. In: The Dynamics of Biological Systems. Math. Planet Earth, vol.4, pp. 189–204. Springer, Cham (2019)
Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. Journal of Functional Analysis. 202(1), 277–305 (2003)
Murray, J.: How the leopard gets its spots. Scientific American 258, 80–87 (1988)
Murray, J.: Mathematical Biology: Spatial Models and Biomedical Applications. Springer, Cham (2003)
Ondreját, M.: Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. (Rozprawy Mat.). 426, 63 (2004)
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations. 26(1–2), 101–174 (2001)
Perthame, B.: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham (2015). Growth, reaction, movement and diffusion
Qiao, H.: A theorem dual to Yamada-Watanabe theorem for stochastic evolution equations. Stoch. Dyn. 10(3), 367–374 (2010)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin-New York (1996)
Schmalfuss, B.: Qualitative properties for the stochastic Navier-Stokes equation. Nonlinear Anal. 28(9), 1545–1563 (1997)
Segel, L.A., Jackson, J.L.: Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–592 (1972)
Sherratt, J.A.: An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51, 183–197 (2005)
Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I. Nonlinearity. 23, 2657–2675 (2010)
Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds. Proc. R. Soc. A 467, 3272–3294 (2011)
Sherratt, J.A., Lord, G.J.: Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. Theor. Popul. Biol. 71, 1–11 (2007)
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79(1), 83–99 (1979)
Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of Bs pq and Fs pq type. Zeitschrift für Analysis und ihre Anwendungen. 14(1), 105–140 (1995)
Simon, J.: Compact sets in the space \(L^p\)(0, T; B). Annali di Matematica Pura ed Applicata. 146(1), 65–96 (1986)
Tessitore, G., Zabczyk, J.: Strict positivity for stochastic heat equations. Stochastic Processes Appl. 77, 83–98 (1998)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam–New York–Oxford (1978)
Turing, A.M.: The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B. 237(641), 37–72 (1952)
Ursino, N.: The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments. Advances in Water Resources 28(9), 956–963 (2005)
van der Stelt, S., Doelman, A., Hek, G., Rademacher, J.D.M.: Rise and fall of periodic patterns for a generalized Klausmeier-Gray-Scott model. Journal of Nonlinear Science. 23(1), 39–95 (2012)
van Neerven, J., Veraar, M., Weis, L.: Stochastic integration in Banach spaces–a survey. In: Stochastic Analysis: a Series of Lectures. Progr. Probab., vol. 68, pp. 297-332. Birkhäuser/Springer, Basel (2015)
Vazquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs. Clarendon Press, Oxford (2007)
Watanabe, S., Yamada, T.: On the uniqueness of solutions of stochastic differential equations. II. J. Math. Kyoto Univ. 11, 553–563 (1971)
Weyl, H.: Über die asymptotische Verteilung der Eigenwerte. Nachr. Ges. Wiss. Göttingen, Math.Phys. Kl. 1911, 110–117 (1911)
Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)
Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36, 1560–1564 (1965)
Woolley, T.E., Baker, R.E., Gaffney, E.A., Maini, P.K.: Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation. Physical Review E. 84(046216), 046216–116 (2011)
Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167 (1971)
Funding
Open Access funding provided by Aalto University. The first author acknowledges partial funding by FWF (Austrian Science Foundation) grant P 28010 “Mathematische Analyse von Flüssigkeitskristallen mitstochastischer Störung”. The second author acknowledges support by the Academy of Finland and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements no. 741487 and no. 818437).
Author information
Authors and Affiliations
Contributions
Both authors contributed equally to the preparation of this manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors have no relevant financial or non-financial interests to disclose. The authors have no competing interests to declare that are relevant to the content of this article.
Ethical Approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Some Useful Inequalities
Lemma A.1
Let \(\gamma >1\), \(x,y\in \mathbb {R}\). Then it holds that
where \(z^{[\gamma ]}:=\vert z\vert ^{\gamma -1}z\) for \(z\in \mathbb {R}\).
Proof
See e.g. [47, Lemma 3.1].
Fix a bounded domain \(\mathcal {O}\subset \mathbb {R}^d\) with sufficiently smooth boundary.
Proposition A.2
For any for \(r\in (2,q+1)\), \(m\in (q+1,\infty )\), \(s\in (0,1)\), with \(\frac{1}{r}\ge \frac{1}{m}-\frac{s}{2}\), and \(\frac{1}{m}\ge \frac{1+s}{q+1}\), there exists a constant \(C>0\) such that
Proof
In order that \(L^m(0,T;H^s_r({{ \mathcal O }}))\) is an interpolation space between
we need that there exists \(\theta \in (0,1)\) such that for the parameters m, r, s the following inequalities are satisfied, see e.g. [3],
Now, if \(\frac{1}{r}\ge \frac{1}{m}-\frac{s}{2}\) and \(\frac{1}{m}\ge \frac{1+s}{q+1}\) are satisfied for \(r\in (2,q+1)\), \(m\in (q+1,\infty )\), \(s\in (0,1)\), then the set of inequalities are satisfied and we obtain Eq. A.2.\(\square \)
Proposition A.3
Let \(s_1,s_2\in \mathbb {R}\) and \(p>1\). Let \(\mathcal {O}\subset \mathbb {R}^d\). Assume that \(s_1\le s_2\) and that \(s_1+s_2>d\left( 0\vee \left( \frac{2}{p}-1\right) \right) \) and \(s_2<\frac{d}{p}\). Then there exists a constant \(C>0\) such that
for any \(r\le s_1+s_2-\frac{d}{p}\) and for any \(u\in H_p^{s_1}(\mathcal {O})\) and any \(v\in H_p^{s_2}(\mathcal {O})\).
Proof
See [58, p. 190, Theorem 1 (iii)].\(\square \)
For the definition of the space \(F^{s}_{p,q}\), we refer to [58, 66, 69]. It translates to classical function spaces as in e.g. [66, Remark 2.1.1], in particular, \(F_{p,2}^{0}=L^p\), \(1<p <\infty \) (Lebesgue spaces), \(F_{p,2}^{m}=W^m_p\), \(m\in \mathbb {N}\) (Sobolev spaces), \(1<p <\infty \) and \(F_{p,2}^{s}= H^s_p\), \(s\in \mathbb {R}\), \(1<p <\infty \) (fractional Sobolev spaces).
Next, we shall record a variant of the Stroock-Varopoulos inequality together with its proof. See e.g. [17, Lemma 3.6] for another version of this result.
Proposition A.4
For any bounded domain \(\mathcal {O}\subset \mathbb {R}^d\) with sufficiently smooth boundary, for any \(T>0\), \(\gamma >1\), \(\theta \in (0,\frac{1}{\gamma })\), there exists a constant \(C=C(\gamma ,T,\mathcal {O})>0\) such that
where \(z^{[\gamma -1]}:=\vert z\vert ^{\gamma -2}z\) for \(z\in \mathbb {R}\).
Proof
By [58, p. 365], we have for any \(p\in (1,\infty )\), \(s\in (0,1)\), \(\mu \in (0,1)\) and \(\varepsilon \in (0,s\mu )\),
From Eq. A.3 we know that for any \(\gamma >1\), \(p\in (1,\infty )\), \(\theta \in (0,\frac{1}{\gamma })\), \(\varepsilon >0\), there exists a constant \(C>0\) such that
In particular, for any \(0<\theta <\frac{1}{\gamma }\) and \(p=2\), there exists a constant \(C>0\) such that
Since we know by the chain rule,
we know that for any \(\theta <\frac{1}{\gamma }\),
Clearly, we have by the fractional Rellich-Kondrachov theorem that for any \(\varepsilon \in (0,\theta )\),
As \(\theta \in (0,\gamma ^{-1})\) is arbitrary, this yields the following result.
Corollary A.5
For any bounded domain \(\mathcal {O}\subset \mathbb {R}^d\) with sufficiently smooth boundary, for any \(T>0\), \(\gamma >1\), \(\theta \in (0,\frac{1}{\gamma })\), there exists a constant \(C=C(\gamma ,T,\mathcal {O})>0\) such that
where \(z^{[\gamma -1]}:=\vert z\vert ^{\gamma -2}z\) for \(z\in \mathbb {R}\).
We have the following embedding.
Proposition A.6
Let \(l_1,l_2\in (2,\infty )\) with \(\frac{d}{2}-\rho \le \frac{2}{l_1}+\frac{d}{l_2}\). Then there exists \( {C} \>{0} \) such that
where \(\mathbb {H}_{\rho }\) is defined in Eq. 3.8.
Proof
First, let us note that for \(\delta >0\) such that
we have the embedding \(H^{\delta }_2({{ \mathcal O }})\hookrightarrow L^{l_2}({{ \mathcal O }})\), and therefore
Due to interpolation, compare with [3, Theorem 5.1.2, p. 107 and Theorem 6.4.5, p. 152], we have
for \(\theta \in (0,1)\) with,
Taking into account that we have
gives that \(\delta \) and \(\theta \) satisfying Eq. A.4 and Eq. A.5 exist and thus the Young inequality for products yields the assertion.\(\square \)
B Function Spaces And The Aubin-Lions-Simon Compactness Theorem
Let B be a separable Banach space, \(0\le c<d<\infty \). Let \(C^{(\beta )}_b(c,d;B)\) denote a set of all continuous and bounded functions \(u:[c,d]\rightarrow B\) such that
is finite. The space \(C_b^{(\beta )}(c,d;E)\) endowed with the norm \(\Vert \cdot \Vert _{C_b^{\beta }(c,d;B)}\) is a Banach space. Let
In addition, for \(1<p <\infty \) let \(W^1_p({{ \mathcal O }})\) be the standard Sobolev space defined by (compare [6, p. 263])
equipped with norm
Given an integer \(m\ge 2\) and a real number \(1\le p<\infty \), we define by induction the space
equipped with norm
Let \(H_2^m({{ \mathcal O }}):=W^m_2({{ \mathcal O }})\), and for \(\rho \in (0,1)\) let \(H_2^\rho ({{ \mathcal O }})\) be the real interpolation space given by \(H^\rho _2({{ \mathcal O }}):=(L^2({{ \mathcal O }}),H^1_2({{ \mathcal O }}))_{\rho ,2}\). In addition, let \(H^{-1}_2({{ \mathcal O }})\) be the dual space of \(H^1_2({{ \mathcal O }})\) and for \(\rho \in (0,1)\) let \(H^{-\rho }_2({{ \mathcal O }})\) be the real interpolation space given by \(H^{-\rho }_2({{ \mathcal O }}):=(L^2({{ \mathcal O }}),H^{-1}_2({{ \mathcal O }}))_{1-\rho ,2}\).
Note, by Theorem 3.7.1 [3], \(H^{-\rho }_2({{ \mathcal O }})\) is dual to \(H^\rho _2({{ \mathcal O }})\), \(\rho \in (0,1)\). Furthermore, we have \((H^{-\rho }_2({{ \mathcal O }}),H^{\rho }_2({{ \mathcal O }}))_{\frac{1}{2},2}=L^2({{ \mathcal O }})\) and \((H^\alpha _2({{ \mathcal O }}),H^\beta _2({{ \mathcal O }}))_{\rho ,2}=H^\theta _2({{ \mathcal O }})\) for \(\theta =\alpha (1-\rho )+\beta \rho \), \(\rho \in (0,1)\) and \(\vert \alpha \vert ,\vert \beta \vert \le 1\).
Since we need it to tackle the compactness, let us introduce the following space. Given \(p\in (1,\infty )\), \(\alpha \in (0,1)\), let \({\mathbb {W}}^ {\alpha }_p (I;B)\) be the Sobolev space of all \(u\in L^p(0,\infty ;B)\) such that
equipped with the norm
Theorem B.1
Let \(B_0\subset B\subset B_1\) be Banach spaces, \(B_0\) and \(B_1\) reflexive, with compact embedding of \(B_0\) to B. Let \(p\in (1,\infty )\) and \(\alpha \in (0,1)\) be given. Let X be the space
Then the embedding of X to \(L^p(0,T;B)\) is compact.
Proof
See [67, p. 86, Corollary 5] or [28, Theorem 2.1].
C On the Burkholder-Davis-Gundy inequality
We collect the exact form of the Burkholder-Davis-Gundy inequality needed here. Given a cylindrical Wiener process W on \(H^\delta _2({{ \mathcal O }})\), \(\delta >1\), over \(\mathfrak {A}=(\Omega ,\mathcal {F},\mathbb {F},{\mathbb {P}})\), and a progressively measurable process \(\xi \in \mathcal {M}^2_\mathfrak {A}(H^\rho _2({{ \mathcal O }}))\), \(\rho \in [0,\frac{1}{2}]\), let us define \(\{Y(t):t\in [0,T]\}\) by
Here, for each \(t\in [0,T]\), \(\xi (t)\) is interpreted as a multiplication operator acting on the elements of \(H^\delta _2(\mathcal {O})\), namely, \(\xi :H_2^\delta (\mathcal {O})\ni \psi \mapsto \xi \psi \in {{ \mathcal S }}'(\mathcal {O})\). Since for any \(\nu >\frac{1}{2}\) and for any \(\varphi \in H^\nu _2({{ \mathcal O }})\) the product \(\xi (t)\varphi \) belongs to \(H^\rho _2({{ \mathcal O }})\) by Proposition A.3, we can view \(\xi (t)\) as a linear map from \(H^\nu _2({{ \mathcal O }})\) into \(H^\rho _2({{ \mathcal O }})\). It is shown in Proposition A.3 that
where the constant \(C>0\) is independent of \(\varphi \). Consequently, for any \(p\ge 1\), \(\delta > 1 \), and any \(\rho \in [0,\frac{1}{2}]\)
where \(\vert \cdot \vert _{L_\text {HS}(H^{\delta }_2,H^{\rho }_2)}\) denotes the Hilbert–Schmidt norm from \(H^{\delta }_2({{ \mathcal O }})\) to \(H^{\rho }_2({{ \mathcal O }})\). First, let us note that for \(\delta >1\) there exists a \(\nu >\frac{1}{2} \) such that the embedding \(H^{\delta }_2({{ \mathcal O }})\hookrightarrow H^\nu _2({{ \mathcal O }})\) is a Hilbert–Schmidt operator. Using the fact that \(\{\psi _k^{(\delta )}:k\in \mathbb {Z} \}\) is an orthonormal basis of \(H^{\delta }_2({{ \mathcal O }})\), and \(\psi _k^{(\delta )}=\lambda _k \psi _k\) we obtain, by using (A.6),
If the embedding \(H^{\delta }_2({{ \mathcal O }}) \hookrightarrow H^{\rho }_2({{ \mathcal O }})\) is supposed to be a Hilbert–Schmidt, the right hand side of Eq. A.7 is finite and we obtain
In case \(\rho \ge \frac{1}{2}\), we use the fact that \(H^\rho _2({{ \mathcal O }})\) is an algebra and obtain
If the embedding \(H^\delta _2({{ \mathcal O }})\hookrightarrow H^{\rho }_2({{ \mathcal O }})\) is a Hilbert–Schmidt operator, then the right hand side of Eq. A.9 is finite and we obtain for \(\delta >\rho +1\)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, 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 changes were made. 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/4.0/.
About this article
Cite this article
Hausenblas, E., Tölle, J.M. The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem. Potential Anal 61, 185–246 (2024). https://doi.org/10.1007/s11118-023-10107-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-023-10107-3
Keywords
- Stochastic Klausmeier evolution system
- Stochastic Schauder-Tychonoff type theorem
- Pattern formation in ecology
- Nonlinear stochastic partial differential equation
- Flows in porous media, pathwise uniqueness