# Tagged Questions

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**0**answers

26 views

### Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...

**1**

vote

**0**answers

110 views

### Runge-Kutta with all nodes at n+1 or zero weights otherwise [migrated]

So, lets say for the family of the explicit Runge-Kutta methods:
$$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$
where,
$$k_1 = hf(t_n, y_n)$$
$$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$
$$\vdots$$
$$k_s = ...

**3**

votes

**2**answers

256 views

### Textbook for Partial Differential Equations with a viewpoint towards Geometry

I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here.
Though similar questions have been asked at ...

**5**

votes

**0**answers

87 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**4**

votes

**1**answer

156 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**4**

votes

**0**answers

131 views

### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...

**4**

votes

**0**answers

155 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**0**

votes

**0**answers

51 views

### Function Related to Jordan Curves

I am looking for a solution to the following problem:
given
a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$
an integrable function $g(s): c(s)\mapsto ...

**1**

vote

**0**answers

97 views

### Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known :
$\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $
$f \rightarrow u$ such that $ - \Delta u = f $
where $ ...

**0**

votes

**0**answers

37 views

### reading request on linear elliptic systems of pdes, strong solutions

Does anyone have some references where I could find results on strong solutions to linear elliptic systems of pdes ?
Regards

**5**

votes

**0**answers

81 views

### $L^p$ regularity for wave equations with coercive boundary conditions

Suppose we have the wave type equation
$$\partial^2_tu - L u = 0$$ on a compact manifold with boundary, where $L$ is a second order strongly elliptic operator with coercive boundary conditions (not ...

**3**

votes

**0**answers

79 views

### Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...

**1**

vote

**0**answers

151 views

### Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty...
My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...

**4**

votes

**1**answer

59 views

### Reference: Hardy space regularity of the Jacobian determinant

I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes.
Theorem:
For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...

**0**

votes

**0**answers

54 views

### References for LWP of a NLS Equation

I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...

**3**

votes

**1**answer

97 views

### Doubling of variables method for parabolic equations

Does anyone have a reference that explains the technique of doubling of variables as introduced by Kruzkov? It seems to be a necessary tool for contraction estimates when we have weak solutions. ...

**1**

vote

**1**answer

89 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 ...

**2**

votes

**2**answers

195 views

### Hardy-Littlewood-Sobolev inequality on hyperbolic space

Let $I_\alpha = (-\Delta)^{-\alpha/2}$ be the Riesz potential on $\mathbb{R}^n$. The Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^n$ says
$$||I_\alpha f||_{L^q} \leq C||f||_{L^p}$$
where $q = ...

**3**

votes

**1**answer

488 views

### Questions on the proof of the Serrin condition for the regularity of Navier-Stokes equations and related issues for the incompressible Euler equation

Edit: The question has been substantially modified from the original one. The original question (see below) concerned with rigorously justifying the proof of the Serrin condition. These questions have ...

**3**

votes

**2**answers

189 views

### Negative real order Sobolev spaces: density and representation

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial ...

**3**

votes

**1**answer

76 views

### reference for existence and blow up results in transport-like PDEs

This question was originally posted by me on math.stackexchange but I didn't get any answers and I thought that perhaps it would be better off here. I hope it's appropriate, I've encountered the ...

**2**

votes

**1**answer

202 views

### Classical theory for the incompressible Euler equation (reference request)

I have recently been interested in the incompressible Euler equation, but since I am new to the topic, I would like to inquire what are the standard sources/references (for self-study) regarding the ...

**0**

votes

**1**answer

122 views

### Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$.
In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...

**3**

votes

**0**answers

65 views

### Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...

**0**

votes

**1**answer

111 views

### Nonlocal (parabolic) PDEs in the Sobolev space setting

Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev ...

**5**

votes

**2**answers

194 views

### Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to
$$
-\Delta\psi = ...

**1**

vote

**0**answers

113 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**0**

votes

**2**answers

135 views

### A basic question about JL Lions' transformation of a Stefan problem

In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...

**1**

vote

**1**answer

203 views

### Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$

Consider the elliptic eigenvalue problem
$$
\begin{cases}
\int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\
\qquad \qquad \qquad \quad ...

**1**

vote

**1**answer

203 views

### Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 ...

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vote

**2**answers

86 views

### References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
...

**0**

votes

**2**answers

145 views

### Defining surface integral on boundary of $C^1$-domain

Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to ...

**1**

vote

**1**answer

122 views

### If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?

Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain.
Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $w'$ means the weak ...

**1**

vote

**1**answer

113 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 ...

**17**

votes

**1**answer

746 views

### The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...

**3**

votes

**0**answers

173 views

### Reference on a Monge-Ampère-like equation

We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$:
$$ \det(Hess(w))=1~, $$
...

**2**

votes

**0**answers

70 views

### What's a good resource for Hormander symbols of type (1/2, 1/2)?

I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...

**3**

votes

**1**answer

235 views

### Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...

**2**

votes

**1**answer

186 views

### Reference request: harnack inequality for distributional solutions of the heat equation

Dear Math Overflowers,
I'm looking for references on the parabolic Harnack inequality for distributional solutions of the heat equation on the whole space
$$
\partial_t u=\Delta u\quad\text{and}\quad ...

**0**

votes

**0**answers

139 views

### Cheeger-Gromov-Taylor theory on manifolds with boundary

I was reading the paper "Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds" by Cheeger, Gromov and Taylor and I am ...

**10**

votes

**1**answer

319 views

### Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...

**0**

votes

**1**answer

121 views

### weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions

I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...

**1**

vote

**2**answers

168 views

### vector valued pde's good reference

I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows.
Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n ...

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vote

**0**answers

44 views

### parabolic PDE with pseudomonotone operators

I am looking for a reference where well-posedness of problems
$$u_t + A(t)u = f$$
is addressed via the Galerkin method where $A$ is a pseudomonotone operator. I am aware that Roubicek's book ...

**2**

votes

**1**answer

248 views

### 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 ...

**3**

votes

**1**answer

195 views

### Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems:
Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...

**1**

vote

**1**answer

143 views

### Nonlinear parabolic PDEs existence with Galerkin method?

Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...

**1**

vote

**1**answer

225 views

### Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...

**1**

vote

**2**answers

142 views

### Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial ...

**3**

votes

**1**answer

202 views

### Heat Equation on $[0,T] \times \mathbb{R}^n$

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.
Do you ...