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

We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: 

$$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x 0, \end{array} \right\quad\quad\quad (\rm CL) $$

where the fluxes fl, fr are mainly assumed to be continuous Developing the ideas of a number of preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers in SIAM J Numer Anal 38(2):681–698, 2000; Towers in SIAM J Numer Anal 39(4):1197–1218, 2001; Towers et al in Skr K Nor Vidensk Selsk 3:1–49, 2003; Adimurthi et al in J Math Kyoto University 43(1):27–70, 2003; Adimurthi et al in J Hyperbolic Differ Equ 2(4):783–837, 2005; Audusse and Perthame in Proc Roy Soc Edinburgh A 135(2):253–265, 2005; Garavello et al in Netw Heterog Media 2:159–179, 2007; Burger et al in SIAM J Numer Anal 47:1684–1712, 2009), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form 

$$ c(x)=c^l11_{\left\{{x 0}\right\}} $$

We refer to such a family as a “germ” It is well known that (CL) admits many different L1 contractive semigroups, some of which reflect different physical applications We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions We devote specific attention to the “vanishing viscosity” germ, which is a way of expressing the “Γ-condition” of Diehl (J Hyperbolic Differ Equ 6(1):127–159, 2009) For any given germ, we formulate “germ-based” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} [in the spirit of Vol’pert (Math USSR Sbornik 2(2):225–267, 1967)] and in the form of a family of global entropy inequalities [following Kruzhkov (Math USSR Sbornik 10(2):217–243, 1970) and Carrillo (Arch Ration Mech Anal 147(4):269–361, 1999)] We characterize those germs that lead to the L1-contraction property for the associated admissible solutions Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities “adapted” to the choice of a germ), or for specific germ-adapted finite volume schemes

/pdf/a-theory-of-l-1-dissipative-solvers-for-scalar-conservation-1i23lfwa38.pdf

A Theory of L 1 -Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

/pdf/a-theory-of-l-1-dissipative-solvers-for-scalar-conservation-3epluv6dhi.pdf

A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux

http://mmns.mimuw.edu.pl/preprints/2011-013.pdf

Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

Scalar conservation laws with general boundary condition and continuous flux function

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)
 t
 − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v
 0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A
 1, A
 2,] with A
 1 ≤ 0 ≤ A
 2 so that the problem is of parabolic-hyperbolic type.

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

Abstract. We study a degenerate scalar conservation law on a bounded domain with non homogeneous boundary condition: b(v)t+div Φ(v) = f on Q := (0, T )×Ω, v(0, ·) = v0 on Ω and v = a on the boundary (0, T )× ∂Ω. The function b is assumed to be continuous nondecreasing and to verify the normalization condition b(0) = 0. Existence and uniqueness of a renormalized entropy solution is established for any Φ ∈ C(R; R ), v0 ∈ L∞(Ω), f ∈ L∞(Q), and boundary data a ∈ L∞(Σ).

/pdf/on-a-degenerate-scalar-conservation-law-with-general-1pek0om2pq.pdf

On a degenerate scalar conservation law with general boundary condition

Renormalized solutions of degenerate elliptic problems

We are interested in the degenerate problem b(v) diva(v,rg(v)) = f with the homogeneous boundary condition g(v) = 0 on some part of the bound- ary. The vector field a is supposed to satisfy the Leray-Lions conditions and the functions b,g to be continuous, nondecreasing and to verify the normal- ization condition b(0) = g(0) = 0 and the range condition R(b + g) = R. Using monotonicity methods, we prove existence and comparison results for renormalized entropy solutions in the L 1 setting.

/pdf/degenerate-stationary-problems-with-homogeneous-boundary-2nn9cdm2x0.pdf

Degenerate stationary problems with homogeneous boundary conditions

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Kaouther Ammar

Author Tools

Chat about Author