scholarly journals A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 852 ◽  
Author(s):  
Jieqiong Shen ◽  
Bin Li
Keyword(s):  
Nonlinear System ◽  
Strong Solution ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Upper Bounds ◽  
Biological Transport ◽  
Transport Networks ◽  
Cross Diffusion ◽  
Priori Estimates

In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

A Priori Estimates for Solutions of "Sub-critical" Equations on CR Sphere

2003 ◽  
Vol 3 (3) ◽  
Author(s):  
J. Prajapat ◽  
Mythily Ramaswamy
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Finite Energy ◽  
Priori Estimates

AbstractHere we study the precise blow-up behaviour and obtain a priori estimates for the finite energy Con the odd dimensional spheres S

scholarly journals Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

2019 ◽  
Vol 39 (2) ◽  
pp. 195-206
Author(s):  
Roberta Filippucci ◽  
Chiara Lini
Keyword(s):  
Existence Theorem ◽  
Liouville Theorem ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Limit Problem ◽  
Gradient Term ◽  
Priori Estimates ◽  
Bounded Smooth ◽  
Quasilinear Problems

In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in \(\mathbb{R}^N\) with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of \(u\) and by removing the growth condition for the nonlinearity \(f\) at \(u=0\).

Uniformly Elliptic Liouville Type Equations Part II: Pointwise Estimates and Location of Blow up Points

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
Daniele Bartolucci ◽  
Luigi Orsina
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Pointwise Estimate ◽  
Divergence Form ◽  
Type Problem ◽  
Pointwise Estimates ◽  
Liouville Type ◽  
Priori Estimates

AbstractWe refine the analysis, initiated in [3], [4] of the blow up phenomenon for the following two dimensional uniformly elliptic Liouville type problem in divergence form:We provide a partial generalization of a result of Y.Y. Li [18] to the case A ≠ I. To this end, in the same spirit of [2], we obtain a sharp pointwise estimate for simple blow up sequences. Moreover, we prove that if {p(∆detA)(pj) = 0, ∀ j = 1, ...,N.This characterization of the blow up set yields an improvement of the a priori estimates already established in [3].

scholarly journals The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Yonghong Wu
Keyword(s):  
Wave Breaking ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Blow Up Rate ◽  
Global Existence Result

The weakly dissipative 2-component Camassa-Holm system is considered. A local well-posedness for the system in Besov spaces is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking mechanisms and the exact blow-up rate of strong solutions to the system are presented. Moreover, a global existence result for strong solutions is derived.

scholarly journals Singularity Formation in the Inviscid Burgers Equation

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 848
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Lower Bound ◽  
Nonlinear Equation ◽  
Burgers Equation ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Singularity Formation ◽  
Priori Estimates ◽  
Inviscid Burgers Equation ◽  
Stability Type

We provide a lower bound for the blow up time of the H2 norm of the entropy solutions of the inviscid Burgers equation in terms of the H2 norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space H2 under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation.

scholarly journals Нелинейная задача для параболического уравнения с неизвестным коэффициентом при производной по времени и ее приложения в математических моделях физико-химических процессов

2017 ◽  
pp. 247-266
Author(s):  
Н.Л. Гольдман
Keyword(s):  
Nonlinear System ◽  
Parabolic System ◽  
A Priori Estimates ◽  
A Priori ◽  
Time Derivative ◽  
Unknown Coefficient ◽  
Smooth Functions ◽  
Priori Estimates ◽  
Physico Chemical ◽  
Rothe Method

Рассматривается нелинейная система с неизвестным коэффициентом при производной по времени в параболическом уравнении и изучаются вопросы существования и единственности ее решения в классе гладких функций. В качестве способа доказательства разрешимости применяется метод прямых Ротэ, который является также и конструктивным методом приближенного решения. Для обоснования метода получены априорные оценки в сеточно-непрерывных классах Гельдера для соответствующей дифференциально-разностной нелинейной системы. Наличие таких оценок позволяет установить сходимость приближенных решений к гладкому решению исходной параболической системы и оценить погрешность метода прямых. Проведенное исследование связано с математическим моделированием физико-химических процессов, в которых происходят изменения внутренних характеристик материалов. Представлен пример задачи о деструкции теплозащитного композиционного материала при высокотемпературном нагреве. We consider conditions of unique solvability in a class of smooth functions for a nonlinear system with an unknown coefficient at the time derivative in a parabolic equation. To this end, the Rothe method is applied, which provides not only the proof of solvability but also the constructive solution of the considered system. A priori estimates in the grid-continuous Holder spaces are established for the corresponding differential-difference nonlinear system that approximates the initial parabolic system by the Rothe method. Such estimates allow one to prove the existence of the smooth solution of this parabolic system and to obtain the error estimates for the Rothe method. This study is connected with the mathematical modelling of physico-chemical processes where the inner characteristics of materials are subjected to changes. As an example, the problem on the destruction of a heat-protective composite under the effect of high-temperature heating is discussed.

scholarly journals Explicit Blowing Up Solutions for a Higher Order Parabolic Equation with Hessian Nonlinearity

2021 ◽  
Author(s):  
Carlos Escudero
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Order Partial Differential Equation ◽  
Infinite Time ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Blowing Up ◽  
Blowing Up Solutions

AbstractIn this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.

A priori bounds and existence of positive solutions to a p-kirchhoff equations

2021 ◽  
pp. 2150082
Author(s):  
Pengfei Li ◽  
Junhui Xie
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Degree Theory ◽  
Kirchhoff Equations ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Dirichlet Boundary Problem

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

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