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14
Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation
, 1995
"... This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: u t + div(vf(u)) = 0 in IR N \Theta [0; T ], with initial data u(:; 0) = u 0 (:) in IR N . where u 0 2 L 1 (IR N ) is a given function, v is a di ..."
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Cited by 40 (6 self)
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This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: u t + div(vf(u)) = 0 in IR N \Theta [0; T ], with initial data u(:; 0) = u 0 (:) in IR N . where u 0 2 L 1 (IR N ) is a given function, v is a divergencefree bounded function of class C 1 from IR N \Theta [0; T ] to IR N , and f is a function of class C 1 from IR to IR. It also gives a result of convergence of a numerical scheme for the discretization of theis equation. We first show the existence of a "process" solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proved; it is also proven that the entropy process solution is in fact a entropy weak solution, hence the existence and uniqueness of the entropy weak solution, and the convergence of the numerical scheme. Keywords...
A Posteriori Error Estimates for Variable TimeStep Discretizations of Nonlinear Evolution Equations
"... We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error ..."
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Cited by 19 (2 self)
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We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive timesteps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.
On A Two Dimensional Elliptic Problem With Large Exponent In Nonlinearity
 Tran. A.M.S
, 1994
"... . A semilinear elliptic equation on a bounded domain in R 2 with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns out that the constrained minimizing problem possesses nice asymptotic behavior as the nonli ..."
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Cited by 11 (5 self)
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. A semilinear elliptic equation on a bounded domain in R 2 with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns out that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that cp , the minimum of energy functional with the nonlinear exponent equal to p, is like (8e) 1=2 p \Gamma1=2 as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded uniformly in p. As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's func...
Asymptotic behavior of an initialboundary value problem for the Vlasov–Poisson–Fokker–Planck system
 SIAM J. Appl. Math
, 1997
"... Abstract. The asymptotic behavior for the Vlasov–Poisson–Fokker–Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the soluti ..."
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Cited by 10 (7 self)
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Abstract. The asymptotic behavior for the Vlasov–Poisson–Fokker–Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson–Boltzmann equation with Dirichlet boundary condition. In the proof of the main result, the conservation law of mass and the balance of energy and entropy identities are rigorously derived. An important argument in the proof is to use a Lyapunovtype functional related to these physical quantities.
Pointwise a posteriori error estimates for monotone semilinear equations
 Numer. Math
"... Abstract. We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semilinear equations. The estimates hold for Lagrange elements of any fixed order, nonsmooth nonlinearities, and take numerical integration into account. The proof hinges o ..."
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Cited by 9 (1 self)
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Abstract. We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semilinear equations. The estimates hold for Lagrange elements of any fixed order, nonsmooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature. 1.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Grenoble 57 no 6
, 2007
"... The paper concerns the magnetic Schrödinger operator H(a, V) = ..."
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Cited by 9 (3 self)
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The paper concerns the magnetic Schrödinger operator H(a, V) =
Extremal solutions and instantaneous complete blowup for elliptic and parabolic problems
, 2005
"... ..."
SOME POSSIBLY DEGENERATE ELLIPTIC PROBLEMS WITH MEASURE DATA AND NON LINEARITY ON THE BOUNDARY
"... Abstract. The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domai ..."
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Abstract. The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates. Résumé. Le but de cet article est l’étude d’équations elliptiques pouvant dégénérer, à données mesures, dans un domaine borné, et avec nonlinéarité au bord du domaine. On étudie deux types de problèmes: un premier est une équation elliptique non dégénérée dans un domaine borné avec des données mesures, supportées à la fois à l’intérieur du domaine et sur le bord de celuici. On traite dans une deuxième partie un problème elliptique dégénéré. On établit des résultat d’existence et de régularité dans les deux cas. Dans les deux problèmes considérés, la nonlinéarité est au bord du domaine.
Contents
, 2006
"... Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density propertie ..."
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Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable CalderónZygmund theory for the problem.