Entropy Solutions for Nonlinear Degenerate Problems
TL;DR: In this article, the existence of entropy solutions for two classes of elliptic-parabolic-hyperbolic degenerate equations with Dirichlet homogeneous boundary conditions was proved.
Abstract: . We consider a class of elliptic‐hyperbolic degenerate equations
$$g(u)-\Delta b(u) +\divg\phi (u) =f$$
with Dirichlet homogeneous boundary conditions and a class of elliptic‐parabolic‐hyperbolic degenerate equations
$$g(u)_t-\Delta b(u) +\divg\phi (u) =f$$
with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function φ satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for φ continuous.
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Book•
26 Oct 2006
TL;DR: The Porous Medium Equation (PME) as discussed by the authors is one of the 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.
Abstract: 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.
978 citations
26 Oct 2006
TL;DR: In this article, the authors introduced the notion of L1-limit solutions for the Dirichlet problem with nonhomogeneous data g 6 = 0 and showed that the L1 norm is a well-defined element of the L∞(Ω) space.
Abstract: 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
766 citations
TL;DR: This work introduces a new offline basis-generation algorithm based on the derivation of rigorous a-posteriori error estimates in various norms for general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations.
Abstract: The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.
420 citations
Cites methods from "Entropy Solutions for Nonlinear Deg..."
...For well-posedness of degenerate parabolic equations with Dirichlet boundary conditions we refer to [6], in the case of mixed Dirichlet-Neumann conditions to [19]....
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TL;DR: A new algorithm is introduced, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way, and it is shown that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions.
Abstract: We present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Frechet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and run-times. In all cases we obtain a good run-time acceleration.
235 citations
Additional excerpts
...[3, 18]....
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TL;DR: A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of $u_{{\mathcal D}}$ to U as the size of the space and time steps tend to zero.
Abstract: One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation
$u_t+{\rm div}({\mathbf q} f(u))-\Delta \phi(u)=0$
by a piecewise constant function
$u_{{\mathcal D}}$
using a discretization
${\mathcal D}$
in space and time and a finite volume scheme. The convergence of
$u_{{\mathcal D}}$
to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on
$u_{{\mathcal D}}$
are used to prove the convergence, up to a subsequence, of
$u_{{\mathcal D}}$
to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of
$u_{{\mathcal D}}$
to{\it u}. Some on a model equation are shown.
221 citations
Cites background or methods from "Entropy Solutions for Nonlinear Deg..."
...This notion has been introduced by several authors ([ 5 ], [20]), who proved the existence of such a solution in bounded domains....
[...]
...For studies of the continuous problem, one can refer to [20], which uses the classical Bardos-Leroux-N´´ elec formulation [1], or [ 5 ] in the case of a homogeneous Dirichlet boundary condition on ∂Ω without condition (4)....
[...]
...In [ 5 ], the existence of a weak solution is proved using semigroup theory (see [2]), and the uniqueness of the entropy weak solution is proved using techniques which have been introduced by S.N....
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References
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Book•
07 Jan 2013
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for
18,443 citations
Book•
01 Jan 1969
7,203 citations
TL;DR: In this paper, a theory of generalized solutions in the large Cauchy's problem for the equations in the class of bounded measurable functions is constructed, and the existence, uniqueness and stability theorems for this solution are proved.
Abstract: In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items.
1,799 citations
"Entropy Solutions for Nonlinear Deg..." refers background or methods in this paper
...It is necessary to introduce Kruzhkov solutions in order to have a good theory of existence and uniqueness (see [Kr1, Kr2 ])....
[...]
...The caseb 0 was solved in R N by Kruzhkov [Kr1, Kr2 ] (see also [BK,...
[...]
TL;DR: In this article, the initial and boundary condition problem for a general first order quasilinear equation in several space variables was solved by using a vanishing viscosity method and gave a definition which chara...
Abstract: We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which chara...
673 citations
Additional excerpts
...Ol]); in a bounded domain it was studied by Hil’debrand [Hi] and by Bardos, Leroux & Nedelec [ BLN ] for smooth data....
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Book•
01 Jan 1972
311 citations