Local well-posedness of 1D degenerate drift diffusion equation

This paper deals with a generalised heat potential for the degenerate (heat) diffusion equation, which satisfies the initial condition with respect to the time variable. An interesting question having several important applications (in general) is what boundary condition can be put on the generalised heat potential on the lateral boundary of the rectangle so that the degenerate diffusion equation complemented by this boundary condition would have a unique solution in the domain still given by the same formula of the generalised heat potential (with the same kernel). This amounts to finding the trace of the generalised heat potential to the lateral boundary of the rectangle. That is in this work a boundary condition for this potential is found. Obtained boundary conditions in the spatial variable will be nonlocal boundary conditions.

Some results on a degenerate and singular diffusion equation

The Euler-Korreweg model results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, which takes into account capillarity effects in regions where the density experiences large variations, typically across interfaces for fluids exhibiting phase changes. One of the main difficulties in the analysis of the Cauchy problem for this model, a third order system of conservation laws, is the absence of dissipative regularization, since viscosity is neglected. The Cauchy problem for isothermal fluids in one space dimension has been addressed by the authors in an earlier paper, using Lagrangian coordinates. Here the Cauchy problem is investigated in arbitrary space dimension N, still for isothermal fluids, and a variable capillarity coefficient. A local well-posedness result is obtained in Sobolev spaces as though the density gradient and the velocity field were solutions of a symmetrizable hyperbolic system. More precisely, well-posedness is shown for Hs+1 x H-s (s > N/2 + 1) perturbations of smooth global solutions, either constant states or traveling profiles. In addition, almost-global existence is proved for small enough perturbations, and a blow-up criterion is shown. Proofs rely on a suitable extended formulation of the system, which turns out to amount to a nonlinear degenerate Schrodinger equation coupled with a transport equation, and on a priori estimates without loss of derivatives for the extended system, which necessitate various 'gauge' functions to cancel out bad commutators.

The Laplacian operator is an isomorphism from s regular functions to s-2 regular functions, for s that is not an integer with inequality controls. When s is an integer it is critical, and the controls need to be modified. This result was extended to the operators L that is degenerate by Lihe Wang (2003). However, the boundary regularity was left undone, and the critical cases are still open. These problems are solved in this thesis by considering the intrinsic distance Holder spaces. Furthermore, the degenerate diffusion equation with nonlocal terms are considered, and Holder regularity is derived.

We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.

Decay of solutions of a degenerate nonlinear diffusion equation

Lototsky-Schnabl operators on the unit interval and degenerate diffusion equations

We study the qualitative properties of degenerate diffusion equations used to describe dispersal processes in population dynamics. For systems of interacting populations, the forms of the diffusion models used determine if the population will intermix or remain disjoint (segregated). The dynamics and stability of segregation boundaries between competing populations is analyzed. General population models with segregation and mixing interactions are derived and connections to behavior in fluid mechanical systems are addressed.

Mathematical analysis of the influence of power-law fluid rheology on a capillary diffusion zone

In this article, we study the initial boundary value problem of the degenerate quasilinear wave equation with a strong dissipation of the form We prove global existence of solutions in Sobolev spaces and general stability estimates using multiplier method and general weighted integral inequalities. Without imposing any growth condition at the origin φ, we show that the energy of the system is bounded above by a quantity, depending on σ and φ, which tends to zero (as time goes to infinity). We also prove the optimality of decay rate of the energy for σ ≡ 1 and φ is slowly degenerates. These estimates allows us to consider large class of functions σ and φ with general growth at the origin. We give many significant examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay.

The free boundary value problem of reaction-diffusion equation is studied. In the first chapter, the Logistic diffusion equation with free boundary and the main work of this paper are introduced. The second chapter is the preparation work, explaining the existence and uniqueness of the solution of the free boundary Logistic diffusion equation. In the third chapter, we prove that messages propagate or disappear in finite time by constructing the asymptotic properties of upper and lower solutions. In chapter 4, the criterion of diffusion disappearance is given, and the threshold of information propagation or disappearance in finite time is given according to the initial value. The results show that when the given value is larger than the threshold value, the information will propagate in the whole region. Otherwise the message will stop propagating for a limited time.

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficient and constant exterior pressure. Under certain assumptions imposed on the initial data, the global existence and uniqueness of a strong solution to FBVP for CNS are established, in particular, the strong solution tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

Multipoint boundary value problems for degenerate differential-operator equations of arbitrary order are studied. Several conditions for the separability in Banach-valued L p -spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular, the sectoriality of this operator is established. As applications, the boundary value problems for degenerate quasielliptic partial differential equations and infinite systems of differential equations on cylindrical domain are studied.