105 research outputs found
Variable Support Control for the Wave Equation: A Multiplier Approach
We study the controllability of the multidimensional wave equation in a
bounded domain with Dirichlet boundary condition, in which the support of the
control is allowed to change over time. The exact controllability is reduced to
the proof of the observability inequality, which is proven by a multiplier
method. Besides our main results, we present some applications
Alternating and variable controls for the wave equation
The present article discusses the exact observability of the wave equation
when the observation subset of the boundary is variable in time. In the
one-dimensional case, we prove an equivalent condition for the exact
observability, which takes into account only the location in time of the
observation. To this end we use Fourier series. Then we investigate the two
specific cases of single exchange of the control position, and of exchange at a
constant rate. In the multi-dimensional case, we analyse sufficient conditions
for the exact observability relying on the multiplier method. In the last
section, the multi-dimensional results are applied to specific settings and
some connections between the one and multi-dimensional case are discussed;
furthermore some open problems are presented.Comment: The original publication is available at www.esaim-cocv.org. The
copyright of this article belongs to ESAIM-COC
Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations
This paper is concerned with the problem of regularization by noise of
systems of reaction-diffusion equations with mass control. It is known that
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 sufficiently noise intensity and spectrum, the blow-up of strong
solutions is delayed and an enhanced diffusion effect is also established.
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 -approach to stochastic PDEs, highlighting new connections
between the two areas.Comment: 50 pages, 1 figure, minor corrections, updated bibliograph
Stochastic maximal -regularity for second order systems with periodic boundary conditions
In this paper we consider an SPDE where the leading term is a second order
systems with periodic boundary conditions with measurable coefficients in
and H\"{o}lder continuous coefficients in space. Assuming
stochastic parabolicity conditions, we prove -estimates. The main novelty is that we do not
require . Moreover, we allow arbitrary and weights
in time. Such mixed regularity estimates play a crucial role in applications to
nonlinear SPDEs which is clear from our previous work, and will be further
applied in forthcoming papers on reaction diffusion equations and Navier-Stokes
equations. 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.Comment: 20 pages, minor correction
Maximal -regularity and -calculus for block operator matrices and applications
Many coupled evolution equations can be described via -block
operator matrices of the form in a product space with possibly unbounded
entries. Here, the case of diagonally dominant block operator matrices is
considered, that is, the case where the full operator can be seen
as a relatively bounded perturbation of its diagonal part with
though with
possibly large relative bound. For such operators the properties of
sectoriality, -sectoriality and the boundedness of the
-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 can be analyzed in
maximal -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.Comment: 55 pages. Accepted for publication in JF
Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity
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 -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 (Agresti and Veraar: Global
existence ... ), the results of this paper are applied in the specific cases of
the Lotka-Volterra equations and the Brusselator model.Comment: accepted for publication in JD
Stability properties of stochastic maximal -regularity
In this paper we consider -regularity estimates for solutions to
stochastic evolution equations, which is called stochastic maximal
-regularity. Our aim is to find a theory which is analogously to Dore's
theory for deterministic evolution equations. He has shown that maximal
-regularity is independent of the length of the time interval, implies
analyticity and exponential stability of the semigroup, is stable under
perturbation and many more properties. We show that the stochastic versions of
these results hold.Comment: reference adde
The critical variational setting for stochastic evolution equations
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. In addition, 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 equation, tamed
Navier-Stokes equations, and Allen-Cahn equation.Comment: This is a minor revision. Accepted for publication in PTR
Nonlinear parabolic stochastic evolution equations in critical spaces part II. Blow-up criteria and instantaneous regularization
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 Pr\"uss-Simonett-Wilke (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 H\"older
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 -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, reaction diffusion equations,
and Allen-Cahn equations. 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.Comment: minor revision. One hypothesis is omitted. One result added in
section
On the trace embedding and its applications to evolution equations
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
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