Nonlinear Elliptic and Parabolic Problems

In the present contribution we study the Stokes operator

A

q

= -

P

q

Δ on

L

σ

q

(Ω), 1 <

q

< ∞, where Ω is a suitable bounded or unbounded domain in ℝ

n

,

n

≥ 2, with

C

1,1

-boundary. We present some conditions on Ω and the related function spaces and basic equations which guarantee that

c

+

A

q

for suitable

c

∈ ℝ is of positive type and admits a bounded

H

∞

- calculus. This implies the existence of bounded imaginary powers of

c

+

A

q

. Most domains studied in the theory of Navier-Stokes like, e.g., bounded, exterior, and aperture domains as well as asymptotically flat layers satisfy the conditions. The proof is done by constructing an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. Finally, the result is used to proof the existence of a bounded

H

∞

-calculus of the Stokes operator Aq on an aperture domain.

The presence of surfactants, ubiquitous at most fluid/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface — so-called Marangoni effects. The mathematical model governing the dynamics of such systems is studied for the case in which the surfactant is soluble in one of the adjacent bulk phases. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk fluids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. Employing

L

p

-maximal regularity we obtain well-posedness of this model for a certain initial configuration. The proof is based on recent

L

p

-theory for two-phase flows without surfactant.

In case of potential perturbations the second resolvent equation transforms the resolvent difference into a product of operators. For obstacle perturbations this behavior is maintained due to Dynkin’s formula. In the present article we study generalized obstacle perturbations, e.g., perturbations by measures with infinite weight. It turns out that the resolvent difference equals a product of two operators, one factor is the free resolvent, the second factor contains all the interaction. This result is applicable to differential operators of arbitrary order and to a wide class of perturbations.

We investigate several aspects of

very weak solutions u

to stationary and nonstationary Navier-Stokes equations in a bounded domain Ω

$$ \subseteq $$

ℝ

3

. This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data

u

|

ϕΩ

=

g

leading to a new and very large solution class. Here we are mainly interested to investigate the ‘largest possible’ class for the more general problem with arbitrary divergence

k

= div

u

, boundary data

g

=

u

|

ϕΩ

. and an external force

f

, as weak as possible. In principle, we will follow Amann’s approach.

We study the asymptotic behavior of positive solutions of a semilinear parabolic equation with a nonlinear boundary condition. This problem admits a unique stationary solution which is not bounded and attracts all positive solutions. We find their growth rate at the singular point on the boundary.

We study initial-boundary value problems for elliptic-parabolic systems of nonlinear partial differential equations describing drift-diffusion- reaction processes of electrically charged species in

N

-dimensional bounded Lipschitzian domains. We include Fermi-Dirac statistics and admit nonsmooth material coefficients. We prove existence and uniqueness of bounded global solutions.

We prove an existence and uniqueness result for an inverse problem arising from a phase-field model with two memory kernels. More precisely, we identify the convolution memory kernels and the diffusion coefficient besides the temperature and the phase-field parameter. We prove our results in the framework of Sobolev spaces. Our fundamental tools are an optimal regularity result in the

L

p

spaces and fixed point arguments.

We are concerned with the appearance of microstructures in some problems of Calculus of Variations experiencing ‘wells’ or minimum of energy. Using piecewise linear finite elements, we give energy estimates and analyze their dependence on the incompatibility of the wells.

In this note we describe recent results on the equations of Navier-Stokes in the exterior of a rotating domain. After rewriting the problem on a fixed exterior domain Ω in ℝ

n

, it is shown that for initial data

u

0

∈

L

σ

p

(Ω) with

p

≥

n

and which are satisfying a certain compatibility condition there exists a unique local mild solution to the Navier-Stokes problem. In the case of the whole space of ℝ

n

, this local mild solution is even analytic in the space variable

x

.

We study properties of the coefficient matrices of non-divergence operators on ℝ

n

aiming at sectoriality and

R

-sectoriality of these operators. In particular, we present results on approximation, scaling, and the behavior in the

L

p

-scale.

A quasilinear degenerate parabolic system modelling the chemotactic movement of cells is studied. The system under consideration has a similar structure as the classical Keller-Segel model, but with the following features: there is a threshold value which the density of cells cannot exceed and the flux of cells vanishes when the density of cells reaches this threshold value. Existence and uniqueness of weak solutions are proved. In the one-dimensional case, flat-hump-shaped stationary solutions are constructed.

The mathematical analysis of this paper shows how the effects of strategic local symbiosis provide an exceptional mechanism to increase productivity in highly competitive environments. The most striking consequence from our analysis is that productivity can blow-up in cooperation areas, though some of the species might become extinct elsewhere, as a result of the aggressions received from competitors. As a by-product, it is realized why strategic symbiosis effects help to avoid massive extinctions of populations, or industrial and financial companies. Going beyond, it has been numerically observed that, in the presence of local strategic symbiosis, high level aggressions might provoke a substantial increment of the complexity of the system; a mechanism that might explain the extraordinary bio-diversity of Earth’s biosphere, as well as the complexity of Global Economy.

We use the degree introduced by P. Benevieri and M. Furi [3] to obtain the generalized minimal cardinal of the number of solutions of the λ- sections of those components of the set of nontrivial solutions of an abstract equation of the form

$$\mathfrak{F}$$

(λ,

x

) = 0 that are compact in one direction of the parameter; here

$$\mathfrak{F}$$

is a

C

1

Fredholm map of index 1 such that

$$\mathfrak{F}$$

(λ, 0) = 0 for all λ ∈ ℝ. These bounds are given in terms of the parity of the linearized Fredholm family

D

2

$$\mathfrak{F}$$

(·, 0). The parity is a local invariant measuring the change of the orientation of

D

2

$$\mathfrak{F}$$

(λ, 0) as λ crosses an interval. The set of eigenvalues of

D

2

$$\mathfrak{F}$$

(·, 0) is not assumed to be discrete. Therefore, as regards applications, the theory developed in this paper should be extremely versatile.

In this work we consider an initial boundary value problem related to the equation

$$u_t - \Delta u + \int_0^t g (t - s)\Delta u(x,s)ds = \left| u \right|^{p - 2} u$$

and prove, under suitable conditions on

g

and

p

, a blow-up result for solutions with negative or vanishing initial energy. This result improves an earlier one by the author.

We describe two finite element algorithms which can be used to study organogenesis or organ development during biological development. Such growth can often be reduced to a free boundary problem with similarities to two-fluid flow in the presence of surface tension, though material is added at a constant growth rate to the developing organ. We use the specific case of avian limb development to discuss our algorithms

We give a simple proof that projecting the 2d Navier-Stokes equations to sufficiently many eigenfunctions of the Stokes operator leads to a system of delayed ODEs. The proof is based on the repeated use of the so-called squeezing property. The reduced system is uniquely solvable and dissipative. Moreover, the solutions on the attractor to the full NSEs are in one-to-one correspondence to the solutions on a compact, invariant subset to a global attractor of the reduced system.

The paper contains a local existence and uniqueness result for quasilinear parabolic equations on a three-dimensional domain including mixed boundary conditions and discontinuous coefficients.

We consider a time-independent BÉnard equation (*) -

Aw

+(λ

0

+ ε)

PMw

+

N

(

u,w

)+(

a

θ

x

+

b

θ

y

)

w

+τ

f

= 0 on the infinite layer

R

2

× [-1/2, 1/2]. Here

f

=

f

(

z

) is an exterior force depending only on

z

∈ [-1/2, 1/2],

w

satisfies Dirichlet conditions in the

z

-direction and

L

1

,

L

2

-periodic conditions in the

x, y

-direction, while

a, b

satisfy a diophantine condition. λ

0

is the critical Rayleigh parameter. It is shown that for generic

f

and small τ, ε,

a, b

, (*) has solutions

w

such that (

a

θ

x

+

b

θ

y

)

w

≠ 0. These solutions give rise to periodic traveling wave solution of the time-dependent version of (*) (with

a

=

b

= 0). The proof is via bifurcation methods related to Hopf bifurcation.

We prove Gagliardo-Nirenberg inequalities for vector-valued Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces Wm p (Rn,E) . Here we do not assume the UMD property for E.

In this paper, we give a survey of results concerning semilinear parabolic problems of the form

u

t

- Δ

u

=

u

p

+

g

(

x, t, u

, ∇

u

). Our goal is to examine the effect of the (gradient) perturbation term g on the asymptotic behavior of blow-up solutions. It turns out that, if the perturbation becomes critical or supercritical in a scaling sense, then the blow-up rate as well as the blow-up profiles may become notably different from those known in the unperturbed case. In some cases, we give precise asymptotic estimates on blow-up solutions.

In this paper a diffusive logistic equation with large diffusion is considered under nonlinear boundary conditions. Non-existence of the corresponding stationary positive solutions is discussed by use of variational techniques.

Continuous coagulation-fragmentation processes with diffusion are studied. It is shown that the parameter dependent diffusion term d(y). generates an analytic semigroup in suitable state spaces even for unbounded diffusion coefficients d(y). This yields existence and uniqueness of local-intime smooth solutions that are global for small initial values in the absence of fragmentation.

We prove

W

p

2,1

(Ω

T

)-estimates (1 <

p

< ∞) for parabolic operators with a second-order elliptic part in non-divergence form with essentially bounded VMO-coefficients. The boundary condition Σ

i

n

=1

b

i

(ξ,

t

)∂

iu

(ξ,

t

) =

g

(ξ,

t

) on ∂Ω

T

is considered in the non-degenerate case, and the

b

i

are only assumed to be in space-time Sobolev-spaces (see condition (

B

)).