Physics of shock waves and high-temperature hydrodynamic phenomena - Ya.B. Zeldovich and Yu.P. Raizer (translated from the Russian and then edited by Wallace D. Hayes and Ronald F. Probstein); Dover Publications, New York, 2002, 944 pp., $34.

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

/pdf/the-porous-medium-equation-mathematical-theory-42juib4cgk.pdf

The Porous Medium Equation: Mathematical Theory

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and

The Porous Medium Equation

Smoothing and Decay Estimates for Nonlinear Diffusion Equations

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszar–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.

/pdf/entropy-dissipation-methods-for-degenerate-parabolic-121cg23bhe.pdf

Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

Extinction behavior for fast diffusion equations with absorption

https://www.math.kth.se/global/preprints/LocalBUFPV06.pdf

Classification of blow-up with nonlinear diffusion and localized reaction

We classify special solutions $u>0$ for the diffusion equation $$ u_t=\left(u^{m-1} u_x\right)_{x}, $$ in the very singular range of parameters, $m\le 0$ (very fast diffusion). We investigate the existence and properties of traveling waves and of self-similar solutions of three types: forward in time, backward in time and exponential type. The study is a necessary preliminary for the construction of a general existence and uniqueness theory of very fast diffusion. We find interesting differences with respect to the standard range $m>0$. In particular, there exist three options in the behaviour at infinity or near a singularity (extending the standard choice between slow and fast rates). The novelty is the existence of solutions with very fast decay as $|x|\to \infty$. There are other ways in which this range differs from usual nonlinear diffusion. Thus, we construct very singular solutions and show that there exists not only one as in the case $m>0$, but infinitely many; moreover, they are classical solution.

/pdf/study-of-self-similarity-for-the-fast-diffusion-equation-3e2yk0vv0z.pdf

Study of self-similarity for the fast-diffusion equation

We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, u t = (u m ) xx (x,t) ∈ R + × (0,T), -(u m ) x (0,t) = u p (0,t) t ∈ (0,T), u(x,0) = u 0 (x) x ∈ R + , where 0 0 are parameters. We describe in terms of p and m when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time T in terms of a self-similar profile.

/pdf/the-blow-up-profile-for-a-fast-diffusion-equation-with-a-3gtos0asik.pdf

The Blow-Up Profile for a Fast Diffusion Equation with a Nonlinear Boundary Condition

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.

/pdf/non-simultaneous-quenching-in-a-system-of-heat-equations-3d3ixrw1ty.pdf

Non-simultaneous quenching in a system of heat equations coupled at the boundary

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