scholarly journals The Elastic Flow with Obstacles: Small Obstacle Results

2021 ◽  
Author(s):  
Marius Müller
Keyword(s):  
Energy Dissipation ◽  
Global Existence ◽  
Elastic Energy ◽  
Obstacle Problem ◽  
Symmetric Cone ◽  
Well Posedness ◽  
Small Obstacle

AbstractWe consider a parabolic obstacle problem for Euler’s elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably ‘small’. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.

scholarly journals Allen–Cahn equation with strong irreversibility

2018 ◽  
Vol 30 (04) ◽  
pp. 707-755
Author(s):  
GORO AKAGI ◽  
MESSOUD EFENDIEV
Keyword(s):  
Energy Dissipation ◽  
Global Attractor ◽  
Comparison Principle ◽  
Obstacle Problem ◽  
Well Posedness ◽  
Smoothing Effect ◽  
Cahn Equation ◽  
Non Linear ◽  
Long Time

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by derivingpartialenergy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solutionu(x,t) converges to a solution of an elliptic obstacle problem ast→ +∞.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

Features of elastic energy dissipation in the PdAgH system

1994 ◽  
Vol 211-212 ◽  
pp. 233-236 ◽  
Author(s):  
F.A. Lewis ◽  
R.-A. McNicholl ◽  
A. Biscarini ◽  
B. Coluzzi ◽  
C. Costa ◽  
...  
Keyword(s):  
Energy Dissipation ◽  
Elastic Energy

scholarly journals Effects of coal's initial macro-cracks on rockburst tendency of rock–coal composite samples

2019 ◽  
Vol 6 (11) ◽  
pp. 181795 ◽  
Author(s):  
Shaojie Chen ◽  
Dawei Yin ◽  
Huimin Liu ◽  
Bing Chen ◽  
Ning Jiang
Keyword(s):  
Energy Dissipation ◽  
Elastic Energy ◽  
Composite System ◽  
Structural Instability ◽  
Uniaxial Loading ◽  
Test Results ◽  
Compression Tests ◽  
Loading Direction ◽  
Attenuation Index ◽  
Composite Samples

In the present study, uniaxial compression tests were conducted on sandstone–coal composite samples to investigate the effects of original macro-cracks in coal on the rockburst tendency. First, the energy dissipation theory was used to derive the elastic energy attenuation index of composite samples during uniaxial loading. Then, based on the test results obtained, the rockburst tendency of composite samples was evaluated and analysed using the uniaxial compressive strength and elastic energy attenuation index. The results show that the original macro-cracks in coal deteriorated the rockburst tendency of composite samples. The original horizontal cracks had the lowest effect on the rockburst tendency, whereas the vertical penetrating cracks through the coal centre (parallel to the loading direction) displayed the greatest effect. The mechanism by which these macro-cracks weakened the rockburst tendency involved two steps: (i) changing the physical properties and energy accumulation conditions of composite samples and (ii) increasing the energy dissipation of composite samples during uniaxial loading. These aspects are important to understand the rockburst hazards induced by the structural instability and failure of the composite system of coal seam and roof rock during deep coal mining.

scholarly journals A diffusion equation with localized chemical reactions

1994 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
John M. Chadam ◽  
Hong-Ming Yin
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Finite Time ◽  
Diffusion Processes ◽  
Model Problem ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Well Posedness ◽  
Blowup Rate ◽  
Time Blowup

In some chemical reaction–diffusion processes, the reaction takes place only at some local sites, due to the presence of a catalyst. In this paper we study the well-posedness of a model problem of this type. Sufficient conditions are found to ensure global existence and finite time blowup. The blowup rate and the blowup set are also investigated in the case of special nonlinearity.

scholarly journals The Effect of Confining Pressure and Water Content on Energy Evolution Characteristics of Sandstone under Stepwise Loading and Unloading

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Shuren Wang ◽  
Paul Hagan ◽  
Yanhai Zhao ◽  
Xu Chang ◽  
Ki-Il Song ◽  
...  
Keyword(s):  
Energy Dissipation ◽  
Water Content ◽  
Elastic Energy ◽  
Strain Energy ◽  
Confining Pressure ◽  
Triaxial Tests ◽  
Energy Evolution ◽  
Evolution Characteristics ◽  
Loading And Unloading ◽  
Stepwise Loading

To investigate the mechanical properties and energy evolution characteristics of sandstone depending on the water contents and confining pressure, the uniaxial and triaxial tests were conducted. The test results show that the strain energy was stored in the sandstone samples at the prepeak stage, and that is suddenly released when the failure occurred, and energy dissipation is sharply increased at the postpeak stage. The damage and energy dissipation characteristics of the samples are observed clearly under the stepwise loading and unloading process. The critical strain energy and energy dissipation show a clear exponential relationship. The critical elastic energy decreases linearly as the water content increases. As the confining pressure increases, the critical elastic energy of the samples transforms from linear to exponential. The concept of energy enhancement factor is proposed to characterize the strengthening effect induced by the confining pressure on the energy storage capacity of the rock samples. The energy evolution of the sandstone samples is more sensitive to the confining pressure than that of the water content.

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