Search

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form u t = a i ′ j ′ u i ′ j ′ + 2 x n γ / 2 a i ′ n u i ′ n + x n γ a nn u nn + b i ′ u i ′ + x n γ / 2 b n u n + c u + f (γ ≤ 1). When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove C s 2 + α -regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity. [ABSTRACT FROM AUTHOR]

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some contracting and convergence results. [ABSTRACT FROM AUTHOR]

Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Cauchy problem of the three-dimensional isentropic compressible Navier–Stokes equations. We prove the global existence and uniqueness of classical solutions with large initial energy and vacuum, under the assumptions that the fluid is nearly isothermal (i.e., the adiabatic exponent γ is sufficiently close to 1), and that the far-field density ρ ~ is either vacuum or close to vacuum. To the best of our knowledge, we establish the first result on the global existence of large-energy solutions with vacuum to the three-dimensional compressible Navier–Stokes equations for the cases of vacuum and nonvacuum far-field constant states, which generalizes the result by Huang, Li and Xin (Commun Pure Appl Math 65:549–585, 2012) on classical solutions with vacuum and small energy (large oscillations). [ABSTRACT FROM AUTHOR]

Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the p-Laplacian type diffusion is also considered. [ABSTRACT FROM AUTHOR]

In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the L p -setting, in particular it is W 2 , p in space. This improves known results in the literature to a large extend. For instance, in Maier and Saal (Discrete Contain Dyn Syst Ser S 7(5):1045–1063, 2014) [Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting W 2 , p regularity, however, is restricted to the interval 1 < p < 1 + δ for small δ > 0 , depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range 1 < p < ∞ . We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every p ∈ (1 , ∞) on the entire L p -space. On the other hand, for the Stokes operator in the space of solenoidal fields L σ p we obtain optimal Sobolev regularity for the full range 1 < p < ∞ and for all opening angles less than π . Roughly speaking, this relies on the fact that an existing "bad" part of L p for the Laplacian is complemented to the space of solenoidal vector fields. [ABSTRACT FROM AUTHOR]

The parabolic-elliptic Keller-Segel system u t = Δ u - ∇ · (u ∇ v) , 0 = Δ v - μ + u , μ : = 1 | Ω | ∫ Ω u , (⋆) is considered under homogeneous Neumann boundary conditions in the ball Ω = B R (0) ⊂ R n . The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of n ≥ 2 and R > 0 there exists a positive number m c = m c (n , R) with the following properties: Whenever m > m c , for any nonconstant nonnegative radial initial data u 0 with ∫ Ω u 0 = m which are, in an appropriately defined sense, more concentrated than the associated spatially homogeneous equilibrium determined by u ≡ m | Ω | , the corresponding initial-value problem for (⋆ ) admits a solution blowing up in finite time; in particular, this implies that any nonconstant and radially nonincreasing initial data u 0 with ∫ Ω u 0 > m c enforce blow-up in (⋆ ). If m < m c , however, then there exist infinitely many nonnegative radial functions u 0 which satisfy ∫ Ω u 0 = m and which are more concentrated than u ≡ m | Ω | , but which yet allow for global bounded solutions to (⋆ ) emanating from u 0 . In consequence, precisely at mass levels above m c the constant steady states of (⋆ ) possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time. [ABSTRACT FROM AUTHOR]

In this paper we obtain sharp Petrovskiĭ criteria for the $$p$$ -parabolic equation, both in the degenerate case $$p>2$$ and the singular case $$1

Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic-parabolic Keller-Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one $$m_c:=2(N-1)/N$$ . In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case. [ABSTRACT FROM AUTHOR]

We prove existence results for the obstacle problem related to the porous medium equation. For sufficiently regular obstacles, we find continuous solutions whose time derivative belongs to the dual of a parabolic Sobolev space. We also employ the notion of weak solutions and show that for more general obstacles, such a weak solution exists. The latter result is a consequence of a stability property of weak solutions with respect to the obstacle. [ABSTRACT FROM AUTHOR]

In this paper we establish the local Hölder continuity of the spatial gradient of weak solutions to the parabolic p( x, t)-Laplacian system More precisely, we prove that provided p(·) and a(·) are Hölder-continuous. [ABSTRACT FROM AUTHOR]

We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence, the uniqueness and the limit in $L^1(\Omega)$ , as $t\to\infty,$ of solutions of general initial-boundary-value problems of the form $\partial_t u - \Delta w= 0$ and $u\in\beta(w)$ in a bounded domain $\Omega$ with dynamical boundary conditions of the form $\partial_t \rho(w)+\partial_\eta w=0.$ [ABSTRACT FROM AUTHOR]

We will show that if u is the solution of the equation $u_t=(\text{log u})_{xx}, u>0$ , in $\Bbb{R}\times(0,T), u(x,0)=u_0(x)\in L^1(\Bbb{R})$ is an even function on $\Bbb{R}$ and is monotone decreasing in $x\ge 0, 0\le u_0\not\equiv 0$ on $\Bbb{R}, \lim_{x\to\infty}\int_a^b(\text{log }u)_x(x,t) dt =-f(t)$ , $\lim_{x\to -\infty}\int_a^b(\text{log }u)_x(x,t) dt=f(t)$ , $\forall 0\int_{\Bbb{R}} u_0 dx$ with $T>0$ being given by $\int_{\Bbb{R}}u_0 dx =2\int_0^Tfdt$ and $f(T)>0$ , then the rescaled function $v(x,s) =u(x,t)/(T-t), s=- \log (T-t)$ , will converge uniformly on every compact subset of $\Bbb{R}$ to $2/(\lambda\text{ cosh}^2 (x/\sqrt{\lambda}))$ as $s\to\infty$ where $\lambda=4/f(T)^2$ . [ABSTRACT FROM AUTHOR]

We introduce a new concept of solution for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in $L^1$ for these solutions. To prove the existence we use the nonlinear semigroup theory. [ABSTRACT FROM AUTHOR]