Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,466 questions
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Regularity for transport equation?
In the book of Evans the transport equation,
$$\frac{d}{dt} u + b\cdot \nabla u = 0, \quad u(t=0)=u_0,$$
is solved by the method of charateristics for $b$ and $u_0$ smooth enogh (in terms of $\mathcal{...
- 151
1
vote
1
answer
449
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Green's function for a certain elliptic equations with rough coefficients
We know the laplacean operator has a Green function which is smooth away from the boundary. Now, consider a linear operator of the form $\partial_i(a^{ij} \partial_j u)$.We can prove that this ...
- 469
2
votes
1
answer
212
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The centralizer of Lienard equation
Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
- 366
0
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0
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Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
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1
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Regularity of the right hand side (the source term) in Evans-Krylov theory
A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...
- 469
3
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0
answers
73
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On the principal eigenvector of an elliptic operator
Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...
- 1,461
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3
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502
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I have this linear PDE...
Hi,
The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$
where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is ...
- 3
2
votes
0
answers
127
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slightly subcritical elliptic pde; the linearized equations
Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems
$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
- 539
2
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1
answer
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A question about the first eigenvalue for two Kahler metrics
While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies $\...
- 303
2
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1
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391
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Fully non-linear PDE
A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
4
votes
3
answers
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Imaginary exponential functional of Brownian motion
Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion:
$X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$
where $g$ is a real scale parameter.
...
- 141
2
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0
answers
87
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1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
4
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2
answers
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Non-Normal derivative boundary conditions for a PDE
For a second order PDE (lets say the Laplace equation), is there a problem with specifying neumann boundary conditions, which instead of being specified in the direction normal to the boundary are ...
- 43
1
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0
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71
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The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...
- 679
2
votes
1
answer
461
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Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?
I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
- 1,471
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vote
0
answers
69
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About Cahn - Hilliard equation solution uniqueness
The uniqueness of the solutions of the Cahn - Hilliard nonlinear PDE
$$\dfrac{\partial c}{\partial t}=\nabla\dot{}(M\nabla\mu)$$ has been proved for many form of the chemical potential $\mu$. What ...
- 399
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0
answers
59
views
$L^{\infty}$ norm of Integral-Differntial equation's solution
Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$
, $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE
$$
\partial_t\Phi(t,z) - \phi(0,z) - \...
- 5,405
2
votes
2
answers
283
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A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
- 21
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1
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Solvability Helmholtz equation [closed]
Let $D=D_1(0)\subset\mathbb{R}^2$ and $\lambda\in\mathbb{R}$, $\lambda>0$.
Consider the Helmholtz operator $L=(\Delta +\lambda I).$
Let $f\in Ker_0(L)$, that is $f$ solves
$$ Lf=0\quad\text{ in $D$...
- 327
1
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0
answers
49
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non-conical support of fundamental solution possible?
In his 1970 paper, on page 124, Hormander discusses fundamental solutions of linear PDE with constant coefficients. I notice he only discusses cases where the support $F$ of the fundamental solution ...
- 1,461
2
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0
answers
82
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Properties of a Sobolev bound
I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...
- 193
1
vote
0
answers
50
views
Verifying general assumption for parabolic PDE
I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon ...
- 187
3
votes
1
answer
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Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?
Hi!
Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type
$$f=1+\mathcal{O}\left( \frac{1}{|z|} \...
- 1,727
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1
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A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$
Let $f_n \to f$ on compact subsets of the real line (these are functions defined on the real line) satisfying some conditions: $f$ has linear growth (but is nonlinear function) and is continuous and ...
- 173
3
votes
1
answer
949
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ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
- 55
2
votes
1
answer
903
views
Weak divergence implies weak differentiability of components?
Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has weak divergence if there exists ...
- 2,222
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0
answers
83
views
Boundedness of a function that satisfies a PDE-type inequality
Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...
1
vote
1
answer
367
views
Getting a comparison principle for parabolic equation when solution is not that smooth
Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to
$$\frac{\partial}{\partial t}b(u) - \Delta u = f$$
where $b$ is continuous, increasing and locally ...
- 145
10
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1
answer
755
views
The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon
In relation to the question on the Hardy inequality and the answer by Terry Tao, I've always been curious about the following:
Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t ...
1
vote
0
answers
186
views
Degeneracy and singularity of the $p$-laplace equation
In what sense is the $p$-Laplacian degenerate for $p$ greater than $2$ and singular for $p$ less than $2$?
- 129
0
votes
1
answer
1k
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Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
1
vote
0
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86
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Wave-like equation with 1st order time derivative and non-constant coefficients
We start with the following recurrence relation for complex coefficients $c_{n,m}$:
$$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m+2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$
where $\dot{c}_{n,m}$...
- 237
-1
votes
1
answer
191
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Nearly elliptic equations [closed]
If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...
- 1
2
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0
answers
154
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Asymptotics of "heat" semigroup
Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...
- 21
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1
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377
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Longtime behaviour of the periodic KdV equation
I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation.
Recall the standard KdV equation ...
- 2,299
4
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0
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231
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Sobolev spaces of maps between manifolds and the Palais-Smale Condition
I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
- 139
5
votes
3
answers
519
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Is there a PDE for this phenomenon?
At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface.
Is there a physical law - like the heat equation - that describes the flow?
Will ...
- 439
3
votes
1
answer
345
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Pseudoinverse of Neumann-Laplacian
Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ~\mathrm{d}\...
- 31
3
votes
0
answers
319
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Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$
Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\...
user21574
4
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1
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A quick and elementary question from Hubbard's Teichmuller Theory : Volume I
Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \...
- 1,471
2
votes
0
answers
112
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Changing frames of the tangent bundle with Schwartz functions [closed]
Let's consider two global frames $\{v_{1},....v_{N}\}$ and $\{u_{1},....u_{N}\}$ of the tangent bundle $T\mathbb{R}^N$.
Now consider the matrix $\{f_{i,j}\}$that change the frame $\{v_k\}$ to $\{u_k\...
- 601
0
votes
1
answer
146
views
Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?
We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:
$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$
where $U(t)= e^{it\Delta} $(free ...
- 1,051
1
vote
0
answers
81
views
About the "method of lines": when are such solutions good approximations for **all** future time?
This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
- 111
2
votes
1
answer
3k
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Brenier's theorem
Hi,
I am reading the book "Topics in Optimal Transportation" by Cedric Villani.
The Brenier's theorem states (among other things) that there is a unique transport plan for the optimal transport with ...
- 481
1
vote
1
answer
265
views
radial limits of subharmonic functions
Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every ...
- 1,701
2
votes
2
answers
656
views
Easy question on Sobolev spaces
I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.
So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
- 1,357
2
votes
2
answers
144
views
First order pde with characteristics [closed]
Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...
- 21
2
votes
1
answer
896
views
Generalized Friedrichs Lemma
Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
- 233
0
votes
1
answer
279
views
Differences between parabolic operators of second order and higher order
Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly ...
- 879
6
votes
0
answers
434
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Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
- 601