# 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.

2,855
questions

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### Uniqueness for a transport-diffusion equation with low integrable drift

Consider the equation
$$
\frac{\partial f}{\partial t} + u \cdot \nabla f - \Delta f = 0
$$
in $(0,T) \times \mathbb R^N$, with initial condition
$$
f \vert_{t=0} = f_0
$$
for some given $f_0 \colon \...

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46 views

### Relation between the $L^2$ norm of the Poisson bracket of $f$ and $g$ and their $H^1$ norms

Let $f,g\in H^1(\Omega)$ where $\Omega$ is a sufficiently nice bounded domain in $\mathbb{R}^2$. If $\{\cdot,\cdot\}:H^1(\Omega)\times H^1(\Omega)\to L^2(\Omega)$ is the Poisson bracket, is there some ...

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66 views

### Does my functional satisfy the Palais Smale condition?

Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...

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134 views

### Nonconstant smooth maps $f:\mathbb{R}^4-\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$

Does there exist a nonconstant smooth map $f:\mathbb{R}^4\setminus\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$ such that each $f(x):\mathbb{R}^4\to\mathbb{R}^5$ is an isometric linear map and $x\...

**2**

votes

**1**answer

96 views

### Gronwall estimate with a Fourier transform

Suppose I have the following equality
$$\hat{u}_\epsilon(t,k) = \alpha(t,k) + \int_0^t\int_{\mathbb{R}^n} e^{ik\cdot(x +\epsilon \phi(s,x))}u_\epsilon(s,x)dxds$$
Where $\alpha(t,k) \geq 0$ and $\alpha(...

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67 views

### Linearization Navier-Stokes

I am considering the classical form of the stationary Navier-Stokes equation given by
$$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...

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14 views

### Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic

I first asked this question on math.stackexchange as I first thought it was somewhat innocent.
I would like if one can solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where
$...

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votes

**1**answer

99 views

### Asymptotics for solution of transport equation and characteristics

Consider the transport equation $$u_t(t,x) + v(t,x) \cdot \nabla u(t,x) = 0.$$
Suppose that the solution of the characteristic equation
$$\dot X(t) = v(t,X(t)) $$
decays to zero as $t \to \infty$. ...

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46 views

### transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$

Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$.
Applying the transformation $\Phi (\psi)=e^{...

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73 views

### Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem

I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...

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30 views

### On maximum principle of spectral fractional Laplacian

Suppose $(-\Delta)^s u=g \geq 0$ in $\Omega$ and $u=0$ in $ \partial \Omega.$ Also suppose $u$ is $C^{2}$ non-negative and $(-\Delta)^s u=0$ in $\Omega \setminus B$ and $u\leq a$ on $\partial B $ ...

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29 views

### Boundary regularity of rectifiable multiplicity 1 hypercurrents

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...

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**2**answers

113 views

### Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation
$$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$
with initial condition $u(0, x) = g(x).$
Suppose that $c(t, x)$ and $...

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votes

**1**answer

187 views

+50

### A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...

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**0**answers

34 views

### Modelling fluid flows with mean curvature flow

A while ago I was wondering if the displacement of fluid described in this blog post
could be modelled with mean curvature flow or some other flow, but when I asked someone in Engineering they replied ...

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40 views

### Harnack inequality for $F(D^{2}u, Du,x)=f$

I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$, ...

**5**

votes

**1**answer

278 views

### When is a distribution having a finite support actually zero?

Let $D$ be a differential operator with smooth coefficients in $\mathbb{R}^n$. Suppose $E$ is a bounded open set in $\mathbb{R}^n$. Suppose $u$ is a function that is smooth up to the boundary of $E$. ...

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46 views

### Complex Monge Ampere equal with zero right hand side

Let $(X, \omega)$ be a compact Kaehler manifold, with $K_{X}$ numerically effective. Suppose that $[\zeta] = c_1(K_X)$ and $\int_{X}\zeta^n = 0$. I am interested in solving
\begin{equation}
(\zeta + i ...

**3**

votes

**1**answer

91 views

### Existence of a special function

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.
Is there any smooth function $...

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**0**answers

51 views

### Itō formula for the solution of a SPDE in the distributional sense

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...

**2**

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**0**answers

48 views

### Modified variational formulation of heat equation

The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to
$$
u_t = \Delta u,
$$
subject to certain boundary conditions and can alternatively be described, in variational form, as ...

**1**

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**1**answer

101 views

### first order derivative of the parabolic equation

Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE
$$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...

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72 views

### Can Schauder's fixed point theorem apply to a metric space?

I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...

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**0**answers

85 views

### Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...

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98 views

### Sobolev extension from a discrete set of points

Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...

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**0**answers

71 views

### Change of one boundary condition in a problem

This is a problem in modeling in hydraulic fracturing field. It's quite long so hopefully someone can patiently read and help me.
The equation numbers are match those of the reference paper by ...

**5**

votes

**0**answers

83 views

### Relations between $\mathcal D$-modules and Exterior Differential Systems?

As a followup of this answer, I wonder relations between $\mathcal D$-modules and Exterior Differential Systems, both of which could be used to describe systems of PDEs. If I understand correctly, $\...

**30**

votes

**2**answers

1k views

### Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where ...

**3**

votes

**1**answer

78 views

### How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by
$$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...

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66 views

### Reference for Rellich Kondrachov theorem on bounded domains and spaces with finite measure

I read at the top of the page $580$ of this paper that the imbedding $W^{1,2}(\Gamma,d\mu) \hookrightarrow L^2(\Gamma,d\mu)$ is compact for a bounded domain $\Gamma \subset \mathbb{R}^n$ and a measure ...

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22 views

### Some scattering properties of cubic NLKG (how can we check the mixed-norm is bounded)?

Let $n=3$ and $u$ be the solution to Klein-Gordon equation
\begin{equation}
\begin{cases}\dot{u}-\Delta u +u=u^3 \\
u(0)=u_0, \partial_t u(0)=u_1,
\end{cases}
\end{equation}
where $(u_0,u_1) \in H^...

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vote

**1**answer

71 views

### A completion equality in product of Sobolev spaces

Let $\Omega$ be a nice bounded domain of $\mathbb{R}^n$. For $s\ge 0$ we define
$$H_{0}^{s}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{H^{s}(\Omega)}.$$
My question: is the following equality true?
$$...

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**0**answers

46 views

### Proof of uniqueness of solution of the Poisson's equation for given boundary conditions [closed]

I asked this question last week on Mathematics Stack Exchange, but I didn't get any feedback. It is not a research level question, however, I post it here, perhaps somebody could help me:
I would ...

**1**

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**0**answers

43 views

### propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...

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**0**answers

44 views

### Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain
$$
\begin{cases}
\rho_t + q_x = 0 \\
q_t + (q^2/\rho + \rho)_x = - q \\
u|_{t=0} = u_0 \\
u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t)
\end{cases}
$$
in which ...

**2**

votes

**1**answer

70 views

### Explicit solution of a free boundary problem for heat equation

Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...

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**0**answers

53 views

### Inf sup condition for discrete Stokes problem

I am considering the discrete inf-sup condition for the discrete Stokes problem
$ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$
This means ...

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**0**answers

84 views

### Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$

Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...

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41 views

### $C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

Consider a Fokker-Planck equation:
$$
\partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0,
$$
with initial condition ...

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**0**answers

31 views

### classical solution of nondegenerate HJB equation

Let $b\in C(\mathbb R)$ and $L \in C_b^2(\mathbb R)$. Consider an equation
$$v_t (x, t) + \inf_{a\in A} \{b(a) v_x(x, t) + a^2 \} + v_{xx}(x, t) + L(x) = 0, \hbox{ on } \mathbb R \times (0, 1)$$
with ...

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66 views

### Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...

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**0**answers

44 views

### Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...

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**0**answers

123 views

### Minimal assumption for an elliptic equation

On the disc $\mathbb{D}$ on the disc with a metric $g=e^{2\lambda} \vert dz \vert^2$(let assume $\lambda$ is smooth on $\overline{\mathbb{D}}$) and I consider either
$$\newcommand{\Div}{\operatorname{...

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72 views

### A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...

**3**

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**0**answers

117 views

### Reference on noncommutative PDE

I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...

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**0**answers

55 views

### Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality:
$$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$
which can be found in many places in the ...

**5**

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**0**answers

120 views

### An inequality for the Hessian of eigenfunctions of the Laplacian on compact manifolds

Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator. Let $\lambda_1 >0$ be the first positive eigenvalue. That is, there exists a non-trivial function ...

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**0**answers

55 views

### Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator

I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why ...

**1**

vote

**1**answer

37 views

### Computing the fractional Laplacian of power function

Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?

**4**

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**1**answer

122 views

### Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753.
Consider the following PDE: ...