scholarly journals Gradient estimates for an orthotropic nonlinear diffusion equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Pierre Bousquet ◽  
Lorenzo Brasco ◽  
Chiara Leone ◽  
Anna Verde
Keyword(s):  
Parabolic Equation ◽  
Diffusion Equation ◽  
Weak Solutions ◽  
Nonlinear Diffusion ◽  
Degenerate Parabolic Equation ◽  
Nonlinear Diffusion Equation ◽  
Gradient Estimates ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Degenerate Parabolic

Abstract We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

scholarly journals On a doubly degenerate parabolic equation with a nonlinear damping term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenbin Xu
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Solution Method ◽  
Nonlinear Damping ◽  
Almost Everywhere ◽  
Viscous Solution ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Diffusion Problems

AbstractConsider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let $v_{\varepsilon }$ v ε be the viscous solution of the equation. By showing that $|\nabla v_{\varepsilon }|\in L^{\infty }(0,T; L_{\mathrm{loc}}^{p(x)}(\Omega ))$ | ∇ v ε | ∈ L ∞ ( 0 , T ; L loc p ( x ) ( Ω ) ) and $\nabla v_{\varepsilon }\rightarrow \nabla v$ ∇ v ε → ∇ v almost everywhere, the existence of weak solutions is proved by the viscous solution method. By imposing some restriction on the nonlinear damping terms, the stability of weak solutions is established. The innovation lies in that the homogeneous boundary value condition is substituted by the condition $a(x)| _{x\in \partial \Omega }=0$ a ( x ) | x ∈ ∂ Ω = 0 , where $a(x)$ a ( x ) is the diffusion coefficient. The difficulties come from the nonlinearity of $\vert {\nabla v} \vert ^{p(x)-2}$ | ∇ v | p ( x ) − 2 as well as the nonlinearity of $|v|^{\alpha (x)}$ | v | α ( x ) .

scholarly journals On a weak solution matching up with the double degenerate parabolic equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Initial Value ◽  
Laplacian Equation ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Regularized Method ◽  
Increasing Function

Abstract The well-posedness of weak solutions to a double degenerate evolutionary $p(x)$ p ( x ) -Laplacian equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)\bigr), $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) ) , is studied. It is assumed that $b(x,t)| _{(x,t)\in \varOmega \times [0,T]}>0$ b ( x , t ) | ( x , t ) ∈ Ω × [ 0 , T ] > 0 but $b(x,t) | _{(x,t)\in \partial \varOmega \times [0,T]}=0$ b ( x , t ) | ( x , t ) ∈ ∂ Ω × [ 0 , T ] = 0 , $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , and $A(s)$ A ( s ) is a strictly monotone increasing function with $A(0)=0$ A ( 0 ) = 0 . A weak solution matching up with the double degenerate parabolic equation is introduced. The existence of weak solution is proved by a parabolically regularized method. The stability theorem of weak solutions is established independent of the boundary value condition. In particular, the initial value condition is satisfied in a wider generality.

scholarly journals On a degenerate parabolic equation from double phase convection

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Double Phase ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$ u t ∈ L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ b ( x ) . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x ∈ ∂ Ω = 0 is found for the first time.

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