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69
Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect ..."
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Cited by 54 (8 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
Existence of Strong Traces for QuasiSolutions of Multidimensional Conservation
 Laws, J. of Hyperbolic Differential Equations
"... Abstract. We consider a conservation law in the domain Ω ⊂ R n+1 with C 1 boundary ∂Ω. For the wide class of functions including generalized entropy sub and supersolutions we prove existence of strong traces for normal components of the entropy fluxes on ∂Ω. No nondegeneracy conditions on the flu ..."
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Cited by 29 (0 self)
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Abstract. We consider a conservation law in the domain Ω ⊂ R n+1 with C 1 boundary ∂Ω. For the wide class of functions including generalized entropy sub and supersolutions we prove existence of strong traces for normal components of the entropy fluxes on ∂Ω. No nondegeneracy conditions on the flux are required. 1
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we u ..."
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Cited by 27 (9 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
Minimal entropy conditions for Burgers equation
 Quarterly Appl. Math. (2004
"... Abstract. We consider strictly convex, 1d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related HamiltonJacobi equation. 1. ..."
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Cited by 10 (1 self)
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Abstract. We consider strictly convex, 1d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related HamiltonJacobi equation. 1.
Diffusive Nwaves and metastability in Burgers equation
"... We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in selfsimilar variables) that captures efficiently the mechanism of transition to the asymptotic states, and allows to estimate the time of evolution from an Nwav ..."
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Cited by 9 (5 self)
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We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in selfsimilar variables) that captures efficiently the mechanism of transition to the asymptotic states, and allows to estimate the time of evolution from an Nwave to the final stage of a diffusion wave. Then, we construct certain special solutions of diffusive Nwaves with unequal masses. Finally, using a set of similarity variables and a variant of the ColeHopf transformation, we obtain an integrated FokkerPlanck equation. The latter is solvable and provides an explicit solution of the viscous Burgers in a series of Hermite polynomials. This format captures the long time  small viscosity interplay, as the diffusion wave and the diffusive Nwaves correspond respectively to the first two terms in the Hermite polynomial expansion.
On the chain rule for the divergence of BV like vector fields: Applications, partial results, open problems
 In Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis. Birkhäuser
, 2006
"... In this paper we study the distributional divergence of vector fields U in R d of the form U = wB, where w is scalar function and B is a weakly differentiable vector field (or more in general the divergence of tensor fields of the form w⊗B). In particular we are interested in a kind of chain rule pr ..."
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Cited by 8 (2 self)
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In this paper we study the distributional divergence of vector fields U in R d of the form U = wB, where w is scalar function and B is a weakly differentiable vector field (or more in general the divergence of tensor fields of the form w⊗B). In particular we are interested in a kind of chain rule property, relating the divergence
Attractors of viscous balance laws: Uniform estimates for the dimension
 J. Di#. Eq
, 1997
"... This paper is devoted to the study of global attractors of a class of singularly perturbed scalar parabolic equations depending on a small parameter ". These equations possess a special structure allowing for a detailed description of the global attractor. Many properties of the attractor can b ..."
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Cited by 7 (3 self)
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This paper is devoted to the study of global attractors of a class of singularly perturbed scalar parabolic equations depending on a small parameter ". These equations possess a special structure allowing for a detailed description of the global attractor. Many properties of the attractor can be deduced using information on equilibria and their variational equations only. This leads to the study of certain singularly perturbed boundary value problems which in general have many solutions. As proposed by Allen and O'Malley [1] for problems where qualitative information on solutions is sought rather than high order approximations we use phase plane methods to describe the solutions of the boundary value problem. As " tends to zero one typically expects that the global attractor has either a very simple structure (e.g. consists of one stable equilibrium only) or that its dimension tends to infinity. The rather surprising result of this paper consists of the proof that for a large class of nonlinearities the dimension of the global attractors stays bounded as " tends to zero. 1.1 Global attractors of scalar parabolic equations
New entropy conditions for scalar conservation law with discontinuous flux
 Discrete and Continuous Dynamical Systems – A; http://arxiv.org/abs/1011.4236
"... Abstract. We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux in rather general form (no convexity or genuine linearity is needed). We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy ..."
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Cited by 6 (4 self)
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Abstract. We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux in rather general form (no convexity or genuine linearity is needed). We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under the assumption that initial data belong to the BVclass. Such initial data enable us to prove that the sequence of solutions to a special vanishing viscosity approximation of the considered equation is, at the same time, the sequence of quasisolutions to a nondegenerate scalar conservation law. This provides existence of the solution admitting strong traces at the interface. The admissibility conditions are chosen so that a kind of crossing condition is satisfied which, together with existence of traces, provides uniqueness of the solution. In the current contribution, we consider the following problem ∂tu + ∂x (H(x)f(u) + H(−x)g(u)) = 0, (t, x) ∈ IR + × IR ut=0 = u0(x) ∈ BV (IR), x ∈ IR (1) where u is the scalar unknown function; u0 is an integrable initial function of bounded variation such that a ≤ u0 ≤ b, a, b ∈ IR; H is the Heaviside function; and
ON A QUASILINEAR DEGENERATE SYSTEM ARISING IN SEMICONDUCTORS THEORY. PART I: EXISTENCE AND UNIQUENESS OF SOLUTIONS∗
"... Abstract. A driftdiffusion model for semiconductors with nonlinear diffusion is considered. The model consists of two quasilinear degenerate parabolic equations for carrier densities and the Poisson equation for electric potential. We assume Lipschitz continuous nonlinearities in the drift and gene ..."
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Cited by 6 (0 self)
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Abstract. A driftdiffusion model for semiconductors with nonlinear diffusion is considered. The model consists of two quasilinear degenerate parabolic equations for carrier densities and the Poisson equation for electric potential. We assume Lipschitz continuous nonlinearities in the drift and generationrecombination terms. Existence of weak solutions is proven by using a regularization technique. Uniqueness of solutions is proven when either the diffusion term ϕ is strictly increasing and solutions have spatial derivatives in L1(QT) or when ϕ is nondecreasing and a suitable entropy condition is fullfilled by the electric potential. Key words. Quasilinear degenerate system, semiconductors. AMS subject classifications. 35K65, 35D05, 35B30, 78A35.
Zero Reaction Limit for Hyperbolic Conservation Laws with Source Terms
, 1998
"... In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term @ t u + @ x f(u) = 1 ffl u(1 \Gamma u 2 ) : For the Cauchy problem to the above equation, we prove that as ffl ! 0, its solution converges to piecewise constant (\Sigma1) solution, where t ..."
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Cited by 5 (4 self)
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In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term @ t u + @ x f(u) = 1 ffl u(1 \Gamma u 2 ) : For the Cauchy problem to the above equation, we prove that as ffl ! 0, its solution converges to piecewise constant (\Sigma1) solution, where the two constants are the two stable local equilibrium. The constants are separated by either shocks that travel with speed 1 2 (f(1) \Gamma f(\Gamma1)), as determined by the RankineHugoniot jump condition, or a nonshock discontinuity that moves with speed f 0 (0), where 0 being the unstable equilibrium. Our analytic tool is the method of generalized characteristics. Similar results for more general source term 1 ffl g(u), having finitely many simple zeros and satisfying ug(u) ! 0 for large juj, are also given. 1 Department of Mathematics, Georgetown University, Washington, DC 20057, USA. Email address: fan@math.georgetown.edu. Research supported in part by NSF grant No. DMS 9705732...