Parabolic Problems

We consider the double obstacle limit for a Navier-Stokes/Cahn- Hilliard type system. The system describes a so-called diffuse interface model for the two-phase flow of two macroscopically immiscible incompressible viscous fluids in the case of matched densities, also known as Model H. Starting with a suitable class of singular free energies, which keep the concentration strictly inside the physically reasonable interval [

a, b

], we analyze a certain singular limit, where the equation for the chemical potential converges to a differential inclusion related to the subgradient of the indicator function of [

a, b

].

The aim of this work is to present a numerical study of generalized Oldroyd-B flows with shear-thinning viscosity in a curved pipe of circular cross section and arbitrary curvature ratio. Flows are driven by a given pressure gradient and behavior of the solutions is discussed with respect to different rheologic and geometric flow parameters.

We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof of maximal regularity for closed and maximal accretive operators following from Kato’s inequality for fractional powers and almost orthogonality arguments.

We study the first and second boundary value problems for parabolic equations in a half-space

$$ \mathbb{R}^{n}_+,\, {n}\geq\,\,{2} $$

, with incompatible initial and boundary data on the boundary

x

n

= 0 of a domain. The existence, uniqueness and estimates of the solutions in the Hölder and weighted spaces are proved. We show that nonfulfillment of the compatibility conditions leads to appearance of the solutions, which are singular in the vicinity of a boundary of a domain as

t

→ 0.

We consider the system of Maxwell-Stefan equations which describe multicomponent diffusive fluxes in non-dilute solutions or gas mixtures. We apply the Perron-Frobenius theorem to the irreducible and quasi-positive matrix which governs the flux-force relations and are able to show normal ellipticity of the associated multicomponent diffusion operator. This provides local-in-time wellposedness of the Maxwell-Stefan multicomponent diffusion system in the isobaric, isothermal case.

We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one-dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.

We give some perturbation theorems for multivalued linear operators in a Banach space. Two different approaches are suggested: the resolvent approach and the modified resolvent approach. The results allow us to handle degenerate abstract Cauchy problems (inclusions). A very wide application of obtained abstract results to initial boundary value problems for degenerate parabolic (elliptic-parabolic) equations with lower-order terms is studied. In particular, integro-differential equations have been considered too.

We consider a semilinear parabolic stochastic integral equation

$$\begin{array}{lll}u(t, \omega, x) = A{a}_a * u(t, \omega, x) + \sum \nolimits ^\infty _{k=1}a_\beta * G^k(t, \omega, u({t, \omega,.}))(x) \\ & + a_\gamma * F(t, \omega, u(t,\omega,.))(x) + u_o(\omega,x) + {t}{u}_1 (\omega, x).\end{array}$$

Here

$$ t \,\epsilon\,[0,\rm T],\omega$$

in a probability space

$$ \Omega,x $$

In a

$$ \sigma $$

-finite measure space

B

with (positive) measure

$$ \Lambda $$

The kernels

$$ a _{\mu}(\rm t) $$

are multiples of

$$ t{\mu-1}. $$

The operator

$$ A :D(A) \subset \rm L_{p}(B)\longrightarrow \rm L_{p}(B)$$

is such that (–A)is a nonnegative operator. The convolution integrals

$$ a\beta\star \rm G^{k}$$

are stochastic convolutions with respect to independent scalar Wiener processes

$$\begin{array}{ll}{w}^{k}\,.\,{F}\,:\,[0,{T}]\,\times\,{\Omega}\,\times{D}((-{A})^\theta)\,\rightarrow \,{L_p}(B)\,{\rm{and}}\,{G}:\,[0,{T}]\,\times\,{\Omega}\,\times{D}((-{A})^\theta)\,\rightarrow \\ \,{L_p}(B,l_2)\end{array} $$

are nonlinear with suitable Lipschitz conditions. We establish an

$${L}_{{p}^-}$$

theory for this equation, including existence and uniqueness of solutions, and regularity results in terms of fractional powers of (

-A

) and fractional derivatives in time.

The global existence of weak solutions is proved for the problem of the motion of several rigid bodies in non-newtonian fluid of power-law with selfgraviting forces.

We consider the periodic

μ

DP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection

∇

on the Fréchet Lie group Diff

∞

(

$$\mathbb{S}^1$$

) of all smooth and orientation-preserving diffeomorphisms of the circle

$$\mathbb{S}^1\,=\,\mathbb{R}/\mathbb{Z}$$

. On the Lie algebra C

∞

(

$$\mathbb{S}^1$$

) of Diff

∞

(

$$\mathbb{S}^1$$

), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of

μ

DP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by

∇

is a smooth local diffeomorphism of a neighbourhood of zero in C

∞

(

$$\mathbb{S}^1$$

) onto a neighbourhood of the unit element in Diff

∞

(

$$\mathbb{S}^1$$

). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C

∞

(

$$\mathbb{S}^1$$

), and a sharp spatial regularity result for the geodesic flow.

In a bounded smooth domain

$$ \Omega \subset \mathbb{R}^{3}\, {\rm {and \,a\,time\,interval}}\, \left[{0}, \,{T}\right.\left)\right.,{0\,<\,{T}\,\leq \,\propto} \, $$

consider the instationary Navier-Stokes equations with initial value

$$ {u}_{o}\,\, \in \,\, {\rm{L}^{2}_{\sigma}(\Omega)\,{and\,external \, force}}{\rm {f}\,=\,{div}\,{F}\,{F}\,\in\,{L}^{2}\,(0,\,T;\,{L}^{2}(\Omega))}$$

As is well known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when

$${u}_{\left|{\delta}{\Omega}\right. }\,\,\,=\,{g}$$

with non-zero time-dependent boundary values

g

. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.

Let

$$ \bar{u}(t) $$

be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let

$$ {z}(t) $$

be the costate associated with

$$ \bar{u}(t) $$

. It is known that integrability of

$$ {z}(t) $$

in the control interval [0, T] guarantees that

$$ \bar{u}(t) $$

is time and norm optimal. However, there are examples where optimality holds (or does not hold) when

$$ {z}(t) $$

is not integrable. This paper presents examples of both cases for a particular semigroup (the right translation semigroup in

$$ {L^2}(0,\infty )$$

).

We consider a body, B, that rotates, without translating, in a Navier-Stokes liquid that fills the whole space exterior to B. We analyze asymptotic properties of steady-state motions, that is, time-independent solutions to the equation of motion written in a frame attached to the body. We prove that “weak” steady-state solutions in the sense of J. Leray that satisfy the energy inequality are Physically Reasonable in the sense of R. Finn, provided the “size” of the data is suitably restricted.

Assuming that the Helmholtz decomposition exists in

$${L}^{q}(\Omega)^{n}$$

it is proved that the Stokes equation has maximal

$${L}^{q}\,{\rm{-regularity\, in\,}}\,{{L}^{s}}_{\sigma}(\Omega)\,{\rm{for}\,s\,\epsilon}\,[min{q,q^\prime}].\,{\rm{Here\,\Omega\,\subset\,\mathbb{R}^n\,is\,an}\,(\varepsilon,\,\propto)}$$

domain with uniform

C

3-boundary.

We consider estimates depending on a parameter for general linear elliptic boundary value problems, with nonhomogeneous boundary conditions, in Nikol’skij spaces. These estimates are then employed to study general linear nonautonomous parabolic systems, again with nonhomogeneous boundary conditions. Maximal regularity results are proved.

In the present paper we study the existence of weak solutions to an abstract parabolic initial-boundary value problem. On the operator appearing in the equation we assume the coercivity conditions given by an

N

-function (i.e., convex function satisfying conditions specified in the paper). The main novelty of the paper consists in the lack of any growth restrictions on the

N

-function combined with an anisotropic character of the

N

-function, namely we allow the dependence on all the directions of the gradient, not only on its absolute value.

We show that elliptic second-order operators

A

of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of

A

are discontinuous and

A

is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

Let

e -tA

be the Stokes semigroup over an unbounded domain O. For construction of the Navier-Stokes flow globally in time, it is crucial to derive

Lq

-

Lr

decay estimate (1.4) for

?e -tA

; thus, given O, we need to ask which (

q, r

) admits (1.4). The present paper provides a principle which interprets how this question is related to spatial decay properties of steady Stokes flow in the domain O under consideration.

This paper deals with global solvability of Westervelt equation, which model arises in the context of high intensity ultrasound. The PDE equations derived are evolutionary quasilinear, potentially degenerate damped wave equations defined on a bounded domain in

Rn

,

n

= 1, 2, 3. We prove local and global well-posedness as well as exponential decay rates for the Westervelt equation with

inhomogeneous Dirichlet boundary conditions

and small Cauchy data. The local existence proofs are based on an application of Banach’s fixed point theorem to an appropriate formulation of these PDEs. Global well-posedness is shown by exploiting functional analytic properties enjoyed by the semigroups generated by strongly damped wave equations. These include analyticity, dissipativity and suitable characterization of fractional powers of the generator – properties that enable the applicability of the “barrier” method. The obtained result holds for all times, provided that the Cauchy data are taken from a suitably small (independent on time) ball characterized by the parameters of the equation, and the boundary data satisfy some decay condition.

We consider second-order divergence form uniformly parabolic and elliptic PDEs with bounded and

VMOx

leading coefficients and possibly linearly growing lower-order coefficients. We look for solutions which are summable to the

p

th power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variables.

We study global regularity properties of transitions kernels associated with second-order differential operators in

R

N

with unbounded drift and potential terms. Under suitable conditions, we prove Sobolev regularity of transition kernels and pointwise upper bounds. As an application, we obtain sufficient conditions implying the differentiability of the associated semigroup on the space of bounded and continuous functions on

R

N

.

The purpose of this paper is to report on recent developments concerning the analysis and the rigorous derivation of thin film models for structures exhibiting residual stress at free equilibria. This phenomenon has been observed in different contexts: growing leaves, torn plastic sheets and specifically engineered polymer gels. The study of wavy patterns in these contexts suggest that the sheet endeavors to reach a non-attainable equilibrium and hence assumes a non-zero stress rest configuration.

We consider a class of second-order linear nonautonomous parabolic equations in ℝ

d

with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator

G

(

t, s

) be compact in

C

b

(ℝ

d

) for

t

>

s

, and describe the asymptotic behavior of

G

(

t, s

)

f

as

t

–

s

→ ∞ in terms of a family of measures

μ

s

,

$$ s \in \mathbb{R} $$

, solution of the associated Fokker-Planck equation.

Let

P

be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem

$$ dU(t) = AU(t)\,dt + dW_{H}(t), $$

where

A

is the generator of a

C

0

-semigroup

S

on a Banach space

E

,

H

is a Hilbert subspace of

E

, and

W

H

is an

H

-cylindrical Brownian motion. Assuming that

S

restricts to a

C

0

-semigroup on

H

, we obtain

L

p

-bounds for

D

H

P

(

t

). We show that if

P

is analytic, then the invariance assumption is fulfilled. As an application we determine the

L

p

-domain of the generator of

P

explicitly in the case where

S

restricts to a

C

0

-semigroup on

H

which is similar to an analytic contraction semigroup. The results are applied to the 1D stochastic heat equation driven by additive space-time white noise.

We prove

$$ \mathcal{R} $$

-sectoriality or, equivalently,

L

p

-maximal regularity for a class of operators on cylindrical domains of the form

$$ \mathbb{R}^{n-k}\times V $$

, where

$$ V \subset \mathbb{R}^{k} $$

is a domain with compact boundary,

$$ \mathbb{R}^{k} $$

, or a half-space. Instead of extensive localization procedures, we present an elegant approach via operatorvalued multiplier theory which takes advantage of the cylindrical shape of both, the domain and the operator.

We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously.

In this paper we investigate quasilinear parabolic systems of conserved Penrose-Fife type. We show maximal

L

p

-regularity for this problem with inhomogeneous boundary data. Furthermore we prove global existence of a solution, under the assumption that the absolute temperature is bounded from below and above. Moreover, we apply the Lojasiewicz-Simon inequality to establish the convergence of solutions to a steady state as time tends to infinity.

We consider a class of singular parabolic problems with Dirichlet boundary conditions. We use the Rayleigh quotient and recent Harnack estimates to derive estimates from above and from below for the solution. Moreover we study the asymptotic behaviour when the solution is approaching the extinction time.

The object of the paper are partial differential delay equations of the form

$$ \dot{u}(t)+Bu(t)\ni\,F(u_{t}),\,t \geq 0,\,u_{0} = \varphi $$

, with

$$ B\,\subset\,X\,\times\,X\,\omega $$

-accretive in a Banach space

X

. We extend the principle of linearized stability around an equilibrium from the semilinear case, with

B

linear, to the fully nonlinear case, with

B

having a linear ‘resolvent-differential’ at the equilibrium.

We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an

L

p

setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces.

In the present note, we address the question about behavior of

L

3

-norm of the velocity field as time

t

approaches blow-up time

T

. It is known that the upper limit of the above norm must be equal to infinity. We show that, for blow-ups of type I, the lower limit of

L

3

-norm equals to infinity as well.

We consider the free boundary problem of the two-phase Navier-Stokes equation with surface tension and gravity in the whole space. We prove a local-in-time unique existence theorem in the space

W

2,1

q,p

with 2 <

p

< ∞ and

n

<

q

< ∞ for any initial data satisfying certain compatibility conditions. Our theorem is proved by the standard fixed point argument based on the maximal

L

p

-

L

q

regularity theorem for the corresponding linearized equations.

The paper contains analysis of the spectrum of a linear problem arising in the study of the stability of a finite isolated mass of uniformly rotating viscous incompressible self-gravitating liquid. It is assumed that the capillary forces on the free boundary of the liquid are not taken into account. It is proved that when the second variation of the energy functional can take negative values, then the spectrum of the problem contains finite number of points with positive real parts, which means instability of the rotating liquid in a linear approximation. The proof relies on the theorem on the invariant subspaces of dissipative operators in the Hilbert space with an indefinite metrics.

We consider a model describing a situation in which a population follows the density gradient of a nutrient that is produced at a spatially inhomogeneous rate of production and subject to diffusion. We show the stability of the equilibrium solution.