31 research outputs found
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Systems of nonlinear PDEs arising in multilayer channel flows
This thesis presents analysis and computations of systems of nonlinear partial differential
equations (PDEs) modelling the dynamics of three stratified immiscible viscous layers
flowing inside a channel with parallel walls inclined to the horizontal. The three layers are
separated by two fluid-fluid interfaces that are free to evolve spatiotemporally and nonlinearly when the flow becomes unstable. The determination of the flow involves solution
of the Navier-Stokes in domains that are changing due to the evolution of the interfaces
whose position must be determined as part of the solution, providing a hard nonlinear
moving boundary problem. Long-wave approximation and a weakly nonlinear analysis
of the Navier-stokes equations along with the associated boundary conditions, leads to
reduced systems of nonlinear PDEs that in general form are systems of coupled Kuramoto-
Sivashinsky equations. These physically derived coupled systems are mathematically rich
due to the rather generic presence of coupled nonlinearities that undergo hyperbolic-elliptic
transitions, along with high order dissipation. Analysis and numerical computations of the
resulting coupled PDEs is presented in order to understand the stability of multilayer channel
flows and explore and quantify the different types of underlying nonlinear phenomena
that are crucial in applications. Importantly, it is found that multilayer flows can be unstable
even at zero Reynolds numbers, in contrast to single interface problems. Furthermore,
the thesis investigates the dynamical behaviour of the zero viscosity limits of the derived
systems in order to verify their physical relevance as reduced models. Strong evidence of
the existence of the zero viscosity limit is provided for mixed hyperbolic-elliptic type systems
whose global existence is an open and challenging mathematical problem. Finally, a
novel sufficient condition is derived for the occurrence of hyperbolic-elliptic transitions in
general conservation laws of mixed type; the condition is demonstrated for several physical systems that have been studied in the literature.Open Acces
Mathematical problems arising in interfacial electrohydrodynamics
In this work we consider the nonlinear stability of thin films in the presence of electric fields. We study a perfectly conducting thin film flow down an inclined plane in the presence of an electric field which is uniform in its undisturbed state, and normal to the plate at infinity. In addition, the effect of normal electric fields on films lying above, or hanging from, horizontal substrates is considered. Systematic asymptotic expansions are used to derive fully nonlinear long wave model equations for the scaled interface motion and corresponding flow fields. For the case of an inclined plane, higher order terms are need to be retained to regularize the problem in the sense that the long wave approximation remains valid for long times. For the case of a horizontal plane the fully nonlinear evolution equation which is derived at the leading order, is asymptotically correct and no regularization procedure is required. In both physical situations, the effect of the electric field is to introduce a non-local term which arises from the potential region above the liquid film, and enters through the electric Maxwell stresses at the interface. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber - surface tension is included and provides a short wavelength cut-off, that is, all sufficiently short waves are linearly stable.
For the case of film flow down an inclined plane, the fully nonlinear equation can produce singular solutions (for certain parameter values) after a finite time, even in the absence of an electric field. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto-Sivashinsky (KS) equation. Global existence and uniqueness results are proved, and refined estimates of the radius of the absorbing ball in L2 are obtained in terms of the parameters of the equations for a generalized class of modified KS equations. The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this, and a general conjecture is made based on extensive computations. We also carry out a complete study of the nonlinear behavior of competing physical mechanisms: long wave instability above a critical Reynolds number, short wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we elucidate parameter regimes that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow, which is known to be stable for this class of problems (an analogous statement holds for horizontally supported films also). Our theoretical results indicate that such highly stable flows can be rendered unstable by using electric fields. This opens the way for possible heat and mass transfer applications which can benefit significantly from interfacial oscillations and interfacial turbulence.
For the case of a horizontal plane, a weakly nonlinear theory is not possible due to the absence of the shear flow generated by the gravitational force along the plate when the latter is inclined. We study the fully nonlinear equation, which in this case is asymptotically correct and is obtained at the leading order. The model equation describes both overlying and hanging films - in the former case gravity is stabilizing while in the latter it is destabilizing. The numerical and theoretical analysis of the fully nonlinear evolution is complicated by the fact that the coefficients of the highest order terms (surface tension in this instance) are nonlinear. We implement a fully implicit two level numerical scheme and perform numerical experiments. We also prove global boundedness of positive periodic smooth solutions, using an appropriate energy functional. This global boundedness result is seenin all our numerical results. Through a combination of analysis and extensive numerical experiments we present evidence for global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena
Pattern formation in the wake of external mechanisms
University of Minnesota Ph.D. dissertation. June 2016. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); xiii, 189 pages.Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws