Cauchy's Problem for Degenerate Second Order Quasilinear Parabolic Equations

Volpert, A. I.; Hudjaev, S I

doi:10.1070/sm1969v007n03abeh001095

Cited by 125 publications

(53 citation statements)

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“…If the initial datum u 0 is bounded in IR N and in BV (IR N ), then solutions of (1.1) can be constructed as in Volpert-Hudjaev [6]. For such initial data, the authors also proved uniqueness.…”

Section: Remarks On the Stabilitymentioning

confidence: 87%

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Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Chen¹,

DiBenedetto

2001

SIAM J. Math. Anal.

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Consider the Cauchy problem for the nonlinear hyperbolic-parabolic equation:where a is a constant vector and u + = max{u, 0}. The equation is hyperbolic in the region [u < 0] and parabolic in the region [u > 0]. It is shown that entropy solutions to (*), that grow at most linearly as |x| → ∞, are stable in a weighted L 1 (IR N ) space, which implies that the solutions are unique. The linear growth as |x| → ∞ imposed on the solutions is shown to be optimal for uniqueness to hold. The same results hold if the Burgers nonlinearity 1 2 au 2 is replaced by a general flux function f (u), provided f (u(x, t)) grows in x at most linearly as |x| → ∞, and/or the degenerate term u + is replaced by a non-decreasing, degenerate, Lipschitz continuous function β(u) defined on IR. For more general β(·), the results continue to hold for bounded solutions.

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Section: Remarks On the Stabilitymentioning

confidence: 87%

“…Both issues are relatively well understood if one removes the hyperbolic part a · ∇u 2 , thereby obtaining a degenerate parabolic equation (see for example the discussion and references of [5,6]). They are equally well understood if one removes the viscosity term ∆u + .…”

Section: Introduction and Resultsmentioning

confidence: 99%

Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Chen¹,

DiBenedetto

2001

SIAM J. Math. Anal.

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“…But neither of these statements settles the question of uniqueness, except for certain special cases, such as the isotropic diffusion, B jk (ρ) = B(ρ)δ jk ≥ 0, e.g., [Ca99], or special cases with mild singularities, e.g., a porous-media type one-point degeneracy, [DiB93,Ta97]. The extension of Kružkov theory to the present context of general parabolic equations with possibly nonisotropic diffusion was completed only recently in [CP03], after the pioneering work [VH69].…”

Section: Nonlinear Degenerate Parabolic Equationsmentioning

confidence: 99%

Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs

Tadmor

Tao

2007

Comm Pure Appl Math

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Abstract. We prove new velocity averaging results for second-order multidimensional equations of the general form,These results quantify the Sobolev regularity of the averages, v f (x, v)φ(v)dv, in terms of the non-degeneracy of the set {v : |L(iξ, v)| ≤ δ} and the mere integrability of the data,Velocity averaging is then used to study the regularizing effect in quasilinear second-order equations, L(∇ x , ρ)ρ = S(ρ) using their underlying kinetic formulations, L(∇ x , v)χ ρ = g S . In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, non-isotropic diffusion.

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“…Hence, it remains to prove (11). For the sake of simplicity, we assume that r(u) takes an algebraic form, namely r(u) = du q + o(u q ) where q > p. Then, by a simple calculation,…”

Section: Lemma 22 Assume That Q(u) Vanishes Algebraically Fast Whenmentioning

confidence: 99%

“…First, if onesigned solutions are uniquely determined by their initial data, Theorem 1.1, it is not known to be true for solutions with changing sign. To this end, entropy conditions are invoked in order to guarantee uniqueness [11]. The two cases differ also in the issue of regularity.…”

Section: Definition 11 a Bounded Function U(x T) Is A Weak Solutiomentioning

confidence: 99%

Regularity of weak solutions of the nonlinear fokker-planck equation

Tassa

1996

Mathematical Research Letters

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Abstract. We study regularity properties of weak solutions of the degenerate parabolic equation. We show that whenever the solution u is nonnegative, Q(u (·, t)) is uniformly Lipschitz continuous and K(u(·, t)) is C 1 -smooth and note that these global regularity results are optimal. Weak solutions with changing sign are proved to possess a weaker regularity -K(u(·, t)), rather than Q(u(·, t)), is uniformly Lipschitz continuous. This regularity is also optimal, as demonstrated by an example due to Barenblatt and Zeldovich.

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Cauchy's Problem for Degenerate Second Order Quasilinear Parabolic Equations

Cited by 125 publications

References 1 publication

Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs

Regularity of weak solutions of the nonlinear fokker-planck equation

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