@article{19505, abstract = {In this paper, we introduce and study the primitive equations with non-isothermal turbulent pressure and transport noise. They are derived from the Navier–Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in H^1 where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.}, author = {Agresti, Antonio and Hieber, Matthias and Hussein, Amru and Saal, Martin}, issn = {1050-5164}, journal = {Annals of Applied Probability}, number = {1}, pages = {635--700}, publisher = {Institute of Mathematical Statistics}, title = {{The stochastic primitive equations with nonisothermal turbulent pressure}}, doi = {10.1214/24-AAP2124}, volume = {35}, year = {2025}, } @article{12485, abstract = {In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. Moreover, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equations, tamed Navier-Stokes equations, and Allen-Cahn equation.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, pages = {957--1015}, publisher = {Springer Nature}, title = {{The critical variational setting for stochastic evolution equations}}, doi = {10.1007/s00440-023-01249-x}, volume = {188}, year = {2024}, } @article{12178, abstract = {In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L² regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.}, author = {Agresti, Antonio and Hieber, Matthias and Hussein, Amru and Saal, Martin}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, keywords = {Applied Mathematics, Modeling and Simulation, Statistics and Probability}, pages = {53--133}, publisher = {Springer Nature}, title = {{The stochastic primitive equations with transport noise and turbulent pressure}}, doi = {10.1007/s40072-022-00277-3}, volume = {12}, year = {2024}, } @article{17372, abstract = {In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for d∈{1,2,3,4} and the Brusselator for d∈{1,2,3}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new Lζ -coercivity/dissipativity conditions and the development of an Lp(Lq) -framework for systems of reaction-diffusion equations, which are needed when treating dimensions d∈{2,3} in the case of cubic or higher order nonlinearities.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, number = {4}, pages = {4870--4927}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems}}, doi = {10.1137/23M1562482}, volume = {56}, year = {2024}, } @article{12486, abstract = {This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the Lp(Lq)-approach to stochastic PDEs, highlighting new connections between the two areas.}, author = {Agresti, Antonio}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, pages = {1907--1981}, publisher = {Springer Nature}, title = {{Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations}}, doi = {10.1007/s40072-023-00319-4}, volume = {12}, year = {2024}, } @article{15098, abstract = {The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.}, author = {Agresti, Antonio and Luongo, Eliseo}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {2727--2766}, publisher = {Springer Nature}, title = {{Global well-posedness and interior regularity of 2D Navier-Stokes equations with stochastic boundary conditions}}, doi = {10.1007/s00208-024-02812-0}, volume = {390}, year = {2024}, } @article{15119, abstract = {In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω) , and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove Lp((0,T)×Ω,tκdt;Hσ,q(Td)) -estimates. The main novelty is that we do not require p=q . Moreover, we allow arbitrary σ∈R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {0246-0203}, journal = {Annales de l'Institut Henri Poincaré, Probabilités et Statistiques}, number = {1}, pages = {413--430}, publisher = {Institute of Mathematical Statistics}, title = {{Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions}}, doi = {10.1214/22-AIHP1333}, volume = {60}, year = {2024}, } @article{12429, abstract = {In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown.}, author = {Agresti, Antonio and Lindemulder, Nick and Veraar, Mark}, issn = {1522-2616}, journal = {Mathematische Nachrichten}, number = {4}, pages = {1319--1350}, publisher = {Wiley}, title = {{On the trace embedding and its applications to evolution equations}}, doi = {10.1002/mana.202100192}, volume = {296}, year = {2023}, } @article{13135, abstract = {In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lp)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1090-2732}, journal = {Journal of Differential Equations}, number = {9}, pages = {247--300}, publisher = {Elsevier}, title = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity}}, doi = {10.1016/j.jde.2023.05.038}, volume = {368}, year = {2023}, } @article{14772, abstract = {Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ A B C D ] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Lpt -regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.}, author = {Agresti, Antonio and Hussein, Amru}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {11}, publisher = {Elsevier}, title = {{Maximal Lp-regularity and H∞-calculus for block operator matrices and applications}}, doi = {10.1016/j.jfa.2023.110146}, volume = {285}, year = {2023}, } @article{11701, abstract = {In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an Lp-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers' equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1361-6544}, journal = {Nonlinearity}, number = {8}, pages = {4100--4210}, publisher = {IOP Publishing}, title = {{Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence}}, doi = {10.1088/1361-6544/abd613}, volume = {35}, year = {2022}, } @article{11858, abstract = {This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical 𝐿2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1424-3202}, journal = {Journal of Evolution Equations}, keywords = {Mathematics (miscellaneous)}, number = {2}, publisher = {Springer Nature}, title = {{Nonlinear parabolic stochastic evolution equations in critical spaces part II}}, doi = {10.1007/s00028-022-00786-7}, volume = {22}, year = {2022}, }