Evolution Equations

These lecture notes are based on  course of Roland from winter semester 2018/19,
though there are small corrections and improvements, as well as minor changes
in the numbering. Typically, the proofs and calculations in the notes are a
bit shorter than those given in class. The drawings and many additional oral
remarks from the lectures are omitted here. On the other hand, the notes
contain very few proofs (of peripheral statements) and a short chapter not
presented during the course. Occasionally I use the notation and definitions of
The lecture notes Analysis 1–4 and Functional Analysis of Roland  without further notice.
I want to thank Roland Schnaubelt for his support and discusion in the preparation of an earlier
version of these notes.

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev-Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed.

L'oggetto principale della prova finale è la teoria degli operatori m-dissipativi su spazi di Banach.
Nella tesi si analizzano alcune proprietà di tale classe di operatori. Il caso più studiato in letteratura è quello degli operatori m-dissipativi definiti su un dominio denso su spazi di Hilbert. Alcuni esempi di operatori m-dissipativi sono forniti dall'operatore Laplaciano con diversi domini di definizione.
Consideriamo ora il problema di evoluzione generale, soffermandoci nei casi particolari in cui A è un operatore autoaggiunto o antisimmetrico in uno spazio di Hilbert.
Nella nostra trattazione introdurremo la teoria dei semigruppi di contrazione, con particolare attenzione al semigruppo di contrazione  generato da un operatore m-dissipativo con dominio denso, analizzando un importante risultato, il Teorema di Hille-Yosida-Phillips, utile nello studio delle equazioni di evoluzione.
In Natura, molti fenomeni di propagazione sono descritti da equazioni di evoluzione, per esempio l'equazione del calore che modellizza il trasporto dell'energia termica in un mezzo omogeneo: essa è il più semplice esempio di equazione di diffuusione che analizziamo nella prova finale come applicazione della teoria dei semigruppi.
La trattazione è basata sui Capitoli 2-3 del libro "An Introduction to Semilinear Evolution Equations" di Thierry Cazenave e Alain Haraux [4].

We extend results of Caffarelli-Silvestre and Stinga-Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C 0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.

Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for $C_0$-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples.

We consider a quite general class of stochastic partial differential equations with quadratic and cubic nonlinearities and derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability. We show that degenerate additive noise has the potential to stabilize or destabilize the dynamics of the dominant modes, due to additional deterministic terms arising in averaging. We focus on equations with quadratic and cubic nonlinearities and give applications to the Burgers' equation, the Ginzburg-Landau equation, and generalized Swift-Hohenberg equation. : 60H15, 60H10

We investigate the stochastic parabolic integral equation of convolution type u = k 1 * A p u + ∞ k=1 k 2 g k + u 0 , t ≥ 0, and develop an L p -theory, 2 ≤ p < ∞, for this equation. The solution u is a function of t, ω, x with ω in a probability space and x ∈ B, a σ-finite measure space with positive measure Λ. The kernels k 1 (t), k 2 (t) are powers of t, i.e., multiples of

We prove here that limits of nonnegative solutions to reaction-diffusion systems whose nonlinearities are bounded in $ L^1 $ always converge to supersolutions of the system. The motivation comes from the question of global existence in time of solutions for the wide class of systems preserving positivity and for which the total mass of the solution is uniformly bounded. We prove that, for a large subclass of these systems, weak solutions exist globally.

We consider stochastic reaction–diffusion equations on a finite network represented by a finite graph. On each edge in the graph, a multiplicative cylindrical Gaussian noise-driven reaction–diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so, we generalize existing results on abstract stochastic reaction–diffusion equations in Banach spaces.

The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles, while for Abresch-Langer type curves, the convergence to m-fold circles happens if and only if the enclosed algebraic area is positive. : Primary 53C44, 35B40; Secondary 35K59, 37B25

Given a star-shaped bounded Lipschitz domain Ω ⊂ R d , we consider the Schrödinger operator L G = −∆ + V on Ω and its restrictions L Ωt G on the subdomains Ωt, t ∈ [0, 1], obtained by shrinking Ω towards its center. We impose either the Dirichlet or quite general Robin-type boundary conditions determined by a subspace G of the boundary space H 1/2 (∂Ω) × H −1/2 (∂Ω), and assume that the potential is smooth and takes values in the set of symmetric (N × N) matrices. Two main results are proved: First, for any t 0 ∈ (0, 1] we give an asymptotic formula for the eigenvalues λ(t) of the operator L Ωt G as t → t 0 up to quadratic terms, that is, we explicitly compute the first and second t-derivatives of the eigenvalues. This includes the case of the eigenvalues with arbitrary multiplicities. Second, we compute the first derivative of the eigenvalues via the (Maslov) crossing form utilized in symplectic topology to define the Arnold-Maslov-Keller index of a path in the set of Lagrangian subspaces of the boundary space. The path is obtained by taking the Dirichlet and Neumann traces of the weak solutions of the eigenvalue problems for L Ωt G .

We prove that the Schrödinger equation defined on a bounded open domain of R n and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L 2 ( ) for any n = 1, 2, 3, . . . , and, moreover, stable on L 2 ( ) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L 2 ( )-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely-at the outset-on a far general result of interest in its own right: an energy estimate at the L 2 ( )-level for a fully general Schrödinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H 1 ( )-level energy estimate to the L 2 ( )-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation. 2000): Primary: 35L05, 35L20. Secondary: 93. -1-0179. I. Lasiecka and R. Triggiani J.evol.equ. I. Lasiecka and R. Triggiani J.evol.equ. u ∈ L 2 (0, T ; L 2 ( 1 )) such that the corresponding solution of the y-problem (1.1) due to the data {y 0 , u} [respectively, due to the data {y 0 = 0, u}] satisfies y(T ) = 0 [respectively, y(T ) = y 1 ]. REMARK 1.2. The above result, Theorem 1.3, complements [L-T.5, Section 8.2], which shows-at least for the Schrödinger's equation on a half-space, in dimension ≥ 2, with Neumann L 2 ( T )-boundary control-that the range of the input-solution operator L T , applied to all of L 2 ( T ), is not contained in H ( ), ∀ > 0.

In a paper by Krylov [6], a parabolic Littlewood-Paley inequality and its application to an L p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations constructed by Krylov [7]. We generalize these inequalities so that they can be applied to stochastic integrodifferential equations. ISBN 978-951-22-9441-1 (print) ISBN 978-951-22-9442-8 (PDF)

Let α be a bounded linear operator in a Banach space X, and let A be a closed operator in this space. Suppose that for 1 , 2 mapping D(A) to another Banach space Y, A | ker 1 and A | ker 2 are generators of strongly continuous semigroups in X. Assume finally that A | ker a , where a = 1 α + 2 β and β = I X − α, is a generator also. In the case where X is an L 1 -type space, and α is an operator of multiplication by a function 0 ≤ α ≤ 1, it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of A | ker 1 with probability α(x) and is subject to the rules of A | ker 2 with probability β(x) = 1 − α(x). We provide an approximation (a singular perturbation) of the semigroup generated by A | ker a by semigroups built from those generated by A | ker 1 and A | ker 2 that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555-565, 2007), Bobrowski and Bogucki (Stud Math 189:287-300, 2008), where semigroups generated by convex combinations of Feller's generators were studied.

Given an admissible measure µ on ∂ where ⊂ R n is an open set, we define a realization µ of the Laplacian in L 2 ( ) with general Robin boundary conditions and we show that µ generates a holomorphic C 0 -semigroup on L 2 ( ) which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form µ . We also show that if D( µ ) contains smooth functions, then µ is of the form dµ = βdσ (where σ is the (n − 1)-dimensional Hausdorff measure and β a positive measurable bounded function on ∂ ); i.e. we have the classical Robin boundary conditions. : 31C15, 31C25, 34D05, 35A15, 35J10, 47D07. Key words: Dirichlet forms, Dirichlet, Neumann and Robin boundary conditions. * This work is part of the DGF-Project: "Regularität und Asymptotik für elliptische und parabolische Probleme" wolfgang arendt and mahamadi warma J.evol.equ.

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphims acting on functions on M in conjunction with maximal regularity and the implicit function theorem.

With advances in technology and the introduction of new materials, the need for new processing methods to enable the benefits of these materials is growing. Liquid crystalline polymers (LCPs) are among a class of high performance polymers but the orientation of crystals in the final product can adversely affect their properties. Modeling the directionality of LCPs during manufacturing processes can provide information and insight into designing new processing methods in order to achieve desired material properties. In this study, a method of modeling the directionality of LCPs for steady state processes such as extrusion is developed. This method can be used for structured and unstructured meshes and as a result is robust to use for complex geometries. To demonstrate this method, a user defined function (UDF) is coded for ANSYS FLUENT and the orientations are shown for 2D cases. Results are shown for Poiseuille flow to verify the alignment of crystals along the shear direction. Results show very close agreement with physical phenomena involved in the rheology of LCPs which include the alignment of crystals in the direction of shear, annihilation of defect structure in quiescence and translation of crystals with the bulk of the fluid. Based on the presented results, the developed method shows very high potential for modeling directionality for steady state processes of LCPs on complex geometries.

Keywords: liquid crystalline polymers, directionality, director, orientation modeling, Frank elastic energy, evolution equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in $${\mathbb{R}^{N}}$$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X j ’s are smooth vector fields in $${\mathbb{R}^{N}}$$, and a ij is a uniformly elliptic matrix. It is assumed that the X j ’s satisfy homogeneity conditions with respect to a group of dilations δ

This paper presents a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is wellposed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper: the reaction-diffusion system is (1) a single linear equation, (2) a FitzHugh-Nagumo system with cubic nonlinearity and (3) a FitzHugh-Nagumo system with a piecewise linear approximation of a cubic nonlinearity. We use a limit argument to obtain solutions in L 1 ∩L ∞. The presented results are a first step in further analysis of these coupled systems: global existence of solutions, positivity, attractors and limit approximations (e.g. diffusion and hydrodynamic limits).

We consider abstract semilinear evolution equations with a time delay feedback. We show that, if the C 0 -semigroup describing the linear part of the model is exponentially stable, then the whole system retains this good property when a suitable smallness condition on the time delay feedback is satisfied. Some examples illustrating our abstract approach are also given.

A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space L 2 (H, ν) where ν is an infinitesimally excessive measure. Then a weak solution is constructed. 2000 Mathematics Subject Classification AMS: 76S05, 35J25, 37L40.

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.

Given a linear semi-bounded symmetric operator S ≥ −ω, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators A Θ of type λ > ω (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type λ) which coincide with the Friedrichs extension of S on a convex set containing the domain of S. The extension parameter Θ ⊂ h × h ranges over the set of nonlinear maximal monotone relations in an auxiliary Hilbert space h isomorphic to the deficiency subspace of S. Moreover A Θ + λ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuous convex function) whenever Θ is sub-potential. Applications to Laplacians with nonlinear singular perturbations supported on null sets and to Laplacians with nonlinear boundary conditions on a bounded set are given.

We study periodic solutions of quasilinear elliptic-parabolic variational inequalities with time-dependent constraints. Assuming that the constraint changes periodically in time, we prove existence of periodic solutions. Moreover, applications of the general results are given.

Given an admissible measure µ on ∂ where ⊂ R n is an open set, we define a realization µ of the Laplacian in L 2 ( ) with general Robin boundary conditions and we show that µ generates a holomorphic C 0 -semigroup on L 2 ( ) which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form µ . We also show that if D( µ ) contains smooth functions, then µ is of the form dµ = βdσ (where σ is the (n − 1)-dimensional Hausdorff measure and β a positive measurable bounded function on ∂ ); i.e. we have the classical Robin boundary conditions. : 31C15, 31C25, 34D05, 35A15, 35J10, 47D07. Key words: Dirichlet forms, Dirichlet, Neumann and Robin boundary conditions. * This work is part of the DGF-Project: "Regularität und Asymptotik für elliptische und parabolische Probleme" wolfgang arendt and mahamadi warma J.evol.equ.

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time-step and the ratio of the space step to the time-step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero whereas the ratio of the space step to the time-step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.

We prove energy estimates for linear p-evolution equations in weighted Sobolev spaces under suitable assumptions on the behavior at infinity of the coefficients with respect to the space variables. As a consequence we obtain well posedness for the related Cauchy problem in the Schwartz spaces S and S ′ .

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier-Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasiliner systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value $0$, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $L_p$-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, wh...

The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper we study the well-posedness for the Cauchy problem associated to this equation in presence of some weak dissipation effects.

Let A be a closed operator on a Banach space X. We study maximal L p-regularity of the problems u (t) = Au(t) + f (t) and u (t) = Au(t) + f (t) on the line. The results are used to solve quasilinear parabolic and elliptic problems on the line.

We obtain an upper estimate for the Poisson kernel for the class of second-order left invariant differential operators on the semi-direct product of the 2n + 1-dimensional Heisenberg group H n and an Abelian group A = R k . We also give an upper estimate for the transition probabilities of the evolution on H n driven by the Brownian motion (with drift) in R k . : 43A85, 31B05, 22E25, 22E30, 60J25, 60J60

Let E be a separable real Banach space and denote by BUC(E) the space of bounded and uniformly continuous functions on E. For a C 0 -semigroup (T (t)) t≥0 acting on BUC(E), we obtain necessary and sufficient conditions ensuring that (T (t)) t≥0 is Gaussian.

We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the clonal evolution model of cancer cell growth. This model has a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. We study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population extinguishes. With telomere restoration, exponential growth occurs in the linear model. We further characterise the specific growth behaviour of the cell population for special cases. We also study the effects of crowding induced mortality on the qualitative behaviour, and the existence and stability of steady states of a nonlinear model incorporating crowding effect. We present examples and extensive numerical simulations, which illustrate the rich dynamic behaviour of the linear and nonlinear models.