QUASISTATIC EVOLUTION IN THE THEORY OF PERFECTLY ELASTO-PLASTIC PLATES PART I: EXISTENCE OF A WEAK SOLUTION

2009 ◽  
Vol 19 (02) ◽  
pp. 229-256 ◽  
Author(s):  
ALEXEY DEMYANOV
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Flow Rule ◽  
Variational Theory ◽  
Absolutely Continuous ◽  
Quasistatic Evolution ◽  
Existence Of Weak Solutions ◽  
Strong Formulation

The existence of weak solutions to the quasistatic problems in the theory of perfectly elasto-plastic plates is studied in the framework of the variational theory for rate-independent processes. Approximate solutions are constructed by means of incremental variational problems in spaces of functions with bounded hessian. The constructed weak solution is shown to be absolutely continuous in time. A strong formulation of the flow rule is obtained.

On weak solutions to a dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases

2020 ◽  
Vol 30 (08) ◽  
pp. 1517-1553
Author(s):  
Young-Sam Kwon ◽  
Antonin Novotny ◽  
C. H. Arthur Cheng
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Open Problem ◽  
Type System ◽  
Convergent Subsequence ◽  
Original System ◽  
Heat Conducting ◽  
Existence Of Weak Solutions ◽  
Essential Step ◽  
Mathematical Literature

In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem.

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

2019 ◽  
Vol 19 (1) ◽  
pp. 197-217 ◽  
Author(s):  
Caifeng Zhang
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Existence Of Weak Solutions ◽  
Laplacian Equation ◽  
Trudinger Inequality ◽  
Dimension One

Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in {W^{s,p}(\mathbb{R}^{N})} . Define \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where {s\in(0,1)} , {sp=N} , {\alpha\in[0,\alpha_{*})} and \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where {V(x)} has a positive lower bound, {f(x,t)} behaves like {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.

DENSITY DEPENDENT STOCHASTIC NAVIER–STOKES EQUATIONS WITH NON-LIPSCHITZ RANDOM FORCING

2010 ◽  
Vol 22 (06) ◽  
pp. 669-697 ◽  
Author(s):  
MAMADOU SANGO
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Initial Density ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Of Weak Solutions ◽  
Density Dependent ◽  
Random Forcing ◽  
Suitable Notion

In this work, we investigate the question of existence of weak solutions to the density dependent stochastic Navier–Stokes equations. The noise considered contains functions which depend nonlinearly on the velocity and which do not satisfy the Lipschitz condition. Furthermore, the initial density is allowed to vanish. We introduce a suitable notion of probabilistic weak solution for the problem and prove its existence.

scholarly journals QUALITATIVE PROPERTIES FOR A SIXTH–ORDER THIN FILM EQUATION

2010 ◽  
Vol 15 (4) ◽  
pp. 457-471 ◽  
Author(s):  
Changchun Liu
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
Approximate Solutions ◽  
Sixth Order ◽  
Classical Solutions ◽  
Qualitative Properties ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Thin Film Equation ◽  
Silicon Based

In this article, the author studies the qualitative properties of weak solutions for a sixth‐order thin film equation, which arises in the industrial application of the isolation oxidation of silicon. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the expansion of the support of solutions are also discussed.

scholarly journals On the Construction of Approximate Solutions for the 1D Pollutant Transport Model

2020 ◽  
Vol 12 (3) ◽  
pp. 1
Author(s):  
Yacouba ZONGO ◽  
Brahima ROAMBA ◽  
Boulaye YIRA
Keyword(s):  
Global Existence ◽  
Weak Solutions ◽  
Transport Model ◽  
Pollutant Transport ◽  
Reynold Number ◽  
Mathematical Structure ◽  
Approximate Solutions ◽  
Global Weak Solutions ◽  
Existence Of Weak Solutions ◽  
The Stability

The purpose of this paper is to build sequences of suitably smooth approximate solutions to the 1D pollutant transport model that preserve the mathematical structure discovered in (Roamba, Zabsonr&eacute;, Zongo, 2017). The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. We show that when the Reynold number goes to infinity, we have always an existence of global weak solutions result for the corresponding model.

scholarly journals Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Changchun Liu ◽  
Junchao Gao ◽  
Songzhe Lian
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximate Solutions ◽  
Divergence Form ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Laplacian Equation

The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs–Thomson law

2011 ◽  
Vol 22 (5) ◽  
pp. 393-422 ◽  
Author(s):  
CHRISTIANE KRAUS
Keyword(s):  
Stefan Problem ◽  
Existence Result ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Existence Of Weak Solutions ◽  
Energy Functionals ◽  
Long Time Existence ◽  
Spatially Inhomogeneous ◽  
Long Time ◽  
Time Existence

The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs–Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law in a weak generalisedBV-formulation.

On the existence of weak solutions for a nonlinear time dependent Dirac equation

1989 ◽  
Vol 113 (1-2) ◽  
pp. 149-158 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Dirac Equation ◽  
Characteristic Function ◽  
Weak Solutions ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Nonlinear Dirac Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisIn this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝwhere X(r) is the characteristic function of a compact interval of ]0, + ∞[

scholarly journals Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

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