arXiv subjects

Helge Kristian Jenssen

Publications and source records attributed to Helge Kristian Jenssen.

At least 19 recordsLinked to original sources

arXiv2026

Cumulative Euler flows

We consider the compressible Euler system for ideal gas flow in the absence of any forces except the internal thermodynamic pressure. In this setting, and in dimensions higher 1, it is known that wave-focusing can drive Euler solutions to amplitude blowup in finite time from bounded initial data. In the known cases (self-similar, radial flows \cites{gud,hun_60,jt3,laz,mrrs1,jls}) the primary flow variables are standard functions at time of blowup. It is natural to ask if the Euler system admits even more singular behavior, and specifically whether accumulation of mass, i.e., the appearance of a Dirac delta in the density field, is possible. We consider the class of radial affine motions \cites{mcvittie,sed, kell,sid_2014} which are conveniently obtained via a Lagrangian formulation. This class does include examples of cumulative behavior, and we observe that there are two distinct mechanisms for accumulation, due to inertial effects or adverse pressure gradients, respectively. However, we show that all affine cumulative solutions necessarily exhibit unphysical behavior due to initially unbounded velocity and/or acceleration in the far-field. We also analyze the behavior of characteristics in cumulative flows and consider concrete examples, including a class of 1-dimensional, non-affine flows. Finally, we discuss the possibility of modifying the known examples to obtain physically acceptable gas flows displaying accumulation.

math.AP↗

arXiv2025

On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$

Explicit irrotational solutions, obtained via the Cole-Hopf transform from the multi-d heat equation, give examples of non-uniqueness for the Cauchy problem in supercritical $L^p$, $W^{1,p}$, and $W^{2,p}$ regimes. We verify non-uniqueness of the trivial solution in the sense of $L^p(\RR^n)$, whenever $n\geq2$ and $1\leq p<n$. The same solutions give non-uniqueness in $W^{1,p}(\RR^n)$ and $W^{2,p}(\RR^n)$ for $1\leq p<\frac{n}{2}$ and $1\leq p<\frac{n}{3}$, respectively. The main example provides solutions which are classical for strictly positive times, and vanish in the stated norms, but explode in $L^\infty(\RR^n)$, as $t\to0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation.

math.AP↗

arXiv2025

Non-isentropic cavity flow for the multi-d compressible Euler system

We rigorously construct non-isentropic and self-similar multi-d Euler flows in which a central cavity (vacuum region) collapses. While isentropic flows of this type have been analyzed earlier by Hunter \cite{hun_60} and others, the non-isentropic setting introduces additional complications, in particular with respect to the behavior along the fluid-vacuum interface. The flows we construct satisfy the physical boundary conditions: the interface is a material surface along which the pressure vanishes, and it propagates with a non-vanishing and finite acceleration until collapse. We introduce a number of algebraic conditions on the parameters in the problem (spatial dimension, adiabatic index, similarity parameters). With these conditions satisfied, a simple argument based on trapping regions for the associated similarity ODEs yields the existence of non-isentropic cavity flows. We finally verify that the conditions are all met for several physically relevant cases in both two and three dimensions.

math.AP↗

arXiv2025

Amplitude blowup in compressible Euler flows without shock formation

Recent works have demonstrated that continuous self-similar radial Euler flows can drive primary (non-differentiated) flow variables to infinity at the center of motion. Among the variables that blow up at collapse is the pressure, and it is unsurprising that this type of behavior can generate an outgoing shock wave. In this work we prove that there is an alternative scenario in which an incoming, continuous 3-d flow suffers blowup, including in pressure, and yet remains continuous beyond collapse. We verify that this behavior is possible even in cases where the fluid is everywhere moving toward the center of motion at time of collapse. The results underscore the subtlety of shock formation in multi-dimensional flow.

math.AP↗

arXiv2023

1-D Isentropic Euler flows: Self-similar Vacuum Solutions

We consider one-dimensional self-similar solutions to the isentropic Euler system when the initial data are at vacuum to the left of the origin. For $x>0$ the initial velocity and sound speed are of form $u_0(x)=u_+x^{1-λ}$ and $c_0(x)=c_+x^{1-λ}$, for constants $u_+\in\RR$, $c_+>0$, $λ\in\RR$. We analyze the resulting solutions in terms of the similarity parameter $λ$, the adiabatic exponent $γ$, and the initial (signed) Mach number $\text{Ma}=u_+/c_+$. Restricting attention to locally bounded data, we find that when the sound speed initially decays to zero in a Hölder manner ($0<λ<1$), the resulting flow is always defined globally. Furthermore, there are three regimes depending on $\text{Ma}$: for sufficiently large positive $\text{Ma}$-values, the solution is continuous and the initial Hölder decay is immediately replaced by $C^1$-decay to vacuum along a stationary vacuum interface; for moderate values of $\text{Ma}$, the solution is again continuous and with an accelerating vacuum interface along which $c^2$ decays linearly to zero (i.e., a "physical singularity''); for sufficiently large negative $\text{Ma}$-values, the solution contains a shock wave emanating from the initial vacuum interface and propagating into the fluid, together with a physical singularity along an accelerating vacuum interface. In contrast, when the sound speed initially decays to zero in a $C^1$ manner ($λ<0$), a global flow exists only for sufficiently large positive values of $\text{Ma}$. Non-existence of global solutions for smaller $\text{Ma}$-values is due to rapid growth of the data at infinity and is unrelated to the presence of a vacuum.

math.AP↗

arXiv2023

Extensions of BV compactness criteria

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fra{ň}kov{á} has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields extensions of the standard compactness criterion for multi-variable functions of bounded variation.

math.FA↗

arXiv2022

Radially Symmetric Non-isentropic Euler flows: continuous blowup with positive pressure

Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions it is therefore unclear if it is the zero-pressure region that is responsible for blowup. The same applies to self-similar Euler flows describing radial cavity flow, first analyzed by Hunter (1960). Recent works have shown that the simplified isothermal and isentropic models admit continuous blowup solutions in the presence of a strictly positive pressure field. In this work we extend this conclusion to the case of the full Euler system. The solutions under consideration are radial self-similar flows in which a continuous wave focuses and blows up. We propagate the solutions beyond blowup and observe numerically that there are cases where an expanding spherical shock wave is generated at collapse. The resulting solution has the unusual property that the flow is isentropic in each of the two regions separated by the shock. We finally verify that these are admissible global weak solutions to the full, multi-d compressible Euler system.

physics.flu-dyn↗

arXiv2022

New Self-similar Euler Flows: gradient catastrophe without shock formation

We consider self-similar solutions to the full compressible Euler system for an ideal gas in two and three space dimensions. The system admits a 2-parameter family of similarity solutions depending on parameters $λ$ and $κ$. Requiring locally finite amounts of mass, momentum, and energy imply certain constraints on $λ$ and $κ$. Further constraints are imposed for particular types of flows. E.g., Guderley's pioneering construction of an unbounded converging shock wave invading a quiescent fluid, requires $κ=0$ and $λ>1$. In this work we analyze the regime $0<λ<1$, which does not appear to have been addressed previously. Our findings include: (i) non-existence of Guderley shock solutions; (ii) existence of bounded and continuous incoming similarity flows in 3-d provided $κ$ takes the value $\hatκ=\frac{2(1-λ)}{γ-1}$, $λ$ is sufficiently small, and $γ$ is sufficiently large; (iii) continuation of the latter flows beyond collapse as globally defined and continuous similarity solutions. A key feature of these solutions is that they, in contrast to Guderley solutions, remain bounded at time of collapse, while the density, velocity, and sound speed all suffer gradient blowup. It is noteworthy that, notwithstanding infinite gradients at collapse, no shock wave appears. The analysis is based on a combination of analytical and numerical calculations.

math.AP↗

arXiv2019

A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$

Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locally prescribed data: $u$ along a non-characteristic curve $M$ and $u_x$ along a non-characteristic curve $N$. We assume that $M$ and $N$ are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the $(x,y)$-plane. It is known that if $M$ is located above $N$, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when $M$ lies below $N$, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function $f$). The construction, via Picard iteration, makes use of a careful choice of additional $u$-data which are updated in each iteration step.

math.AP↗

arXiv2019

Multi-d Isothermal Euler Flow: Existence of unbounded radial similarity solutions

We show that the multi-dimensional compressible Euler system for isothermal flow of an ideal, polytropic gas admits global-in-time, radially symmetric solutions with unbounded amplitudes due to wave focusing. The examples are similarity solutions and involve a converging wave focusing at the origin. At time of collapse, the density, but not the velocity, becomes unbounded, resulting in an expanding shock wave. The solutions are constructed as functions of radial distance to the origin $r$ and time $t$. We verify that they provide genuine, weak solutions to the original, multi-d, isothermal Euler system. While motivated by the well-known Guderley solutions to the full Euler system for an ideal gas, the solutions we consider are of a different type. In Guderley solutions an incoming shock propagates toward the origin by penetrating a stationary and "cold" gas at zero pressure (there is no counter pressure due to vanishing temperature near the origin), accompanied by blowup of velocity and pressure, but not of density, at collapse. It is currently not known whether the full system admits unbounded solutions in the absence of zero-pressure regions. The present work shows that the simplified isothermal model does admit such behavior.

math.AP↗

arXiv2018

On similarity flows for the compressible Euler system

Radial similarity flow offers a rare instance where concrete inviscid, multi-dimensional, compressible flows can be studied in detail. In particular, there are flows of this type that exhibit imploding shocks and cavities. In such flows the primary flow variables (density, velocity, pressure, temperature) become unbounded at time of collapse. In both cases the solution can be propagated beyond collapse by having an expanding shock wave reflect off the center of motion. These types of flows are of relevance in bomb-making and inertial confinement fusion, and also as benchmarks for computational codes; they have been investigated extensively in the applied literature. However, despite their obvious theoretical interest as examples of unbounded solutions to the multi-dimensional Euler system, the existing literature does not address to what extent such solutions are bona fide weak solutions. In this work we review the construction of globally defined radial similarity shock and cavity flows, and give a detailed description of their behavior following collapse. We then prove that similarity shock solutions provide genuine weak solutions, of unbounded amplitude, to the multi-dimensional Euler system. However, both types of similarity flows involve regions of vanishing pressure prior to collapse (due to vanishing temperature and vacuum, respectively) - raising the possibility that Euler flows may remain bounded in the absence of such regions.

math.AP↗

arXiv2018

A generalization of an integrability theorem of Darboux

In his monograph "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Principes de géométrie analytique", 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type \[\partial_{x_i} u_α(x)=f^α_i(x,u(x)),\quad i\in I_α\subseteq\{1,\dots,n\}.\] For a given point $\bar x\in \mathbb{R}^n$ it is assumed that the values of the unknown $u_α$ are given locally near $\bar x$ along $\{x\,|\, x_i=\bar x_i \, \text{for each}\, i\in I_α\}$. The more general of the theorems, Théorème III, was proved by Darboux only for the cases $n=2$ and $3$. In this work we formulate and prove a generalization of Darboux's Théorème III which applies to systems of the form \[{\mathbf r}_i(u_α)\big|_x = f_i^α(x, u(x)), \quad i\in I_α\subseteq\{1,\dots,n\}\] where $\mathcal R=\{{\mathbf r}_i\}_{i=1}^n$ is a fixed local frame of vector fields near $\bar x$. The data for $u_α$ are prescribed along a manifold $Ξ_α$ containing $\bar x$ and transverse to the vector fields $\{{\mathbf r}_i\,|\, i\in I_α\}$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $\mathcal R$ and on the manifolds $Ξ_α$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $C^1$-solution via Picard iteration for any number of independent variables $n$.

math.AP↗

arXiv2017

On Two Theorems of Darboux

We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation contains a single derivative for which it is solved: \[\frac{\partial u_i}{\partial x_j}(x)=f_{ij}(x,u(x)).\] The data prescribe values for the unknowns $u_i$ along certain hyperplanes through a given point $\bar x$. The first theorem applies to determined systems (the number of equations equals the number unknowns), and a unique, local solution is obtained via Picard iteration. While Darboux's statement of the theorem leaves unspecified "certaines conditions de continuité," it is clear from his proof that he assumes Lipschitz continuity of the maps $f_{ij}$. On the other hand, he did not address the regularity of the data. We provide a precise formulation and proof of his first theorem. The second theorem is more involved and applies to overdetermined systems of the same general form. Under the appropriate integrability conditions, Darboux used his first theorem to treat the cases with two and three independent variables. We provide a proof for any number of independent variables. While the systems are rather special, they do appear in applications; e.g., the second theorem contains the classical Frobenius theorem on overdetermined systems as a special case. The key aspect of the proofs is that they apply to non-analytic situations. In an analytic setup the results are covered by the general Cartan-Kähler theorem.

math.AP↗

arXiv2017

Jacobians with prescribed eigenvectors

Let $Ω\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $Ω$, that is a set of $m$ linearly independent vector fields prescribed on $Ω$ ($m\leq n$). We consider the issue of describing the set of all maps $F:Ω\to\mathbb{R}^n$ with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of $F$. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame, in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich, can be completely analyzed. Involutive, but non-rich case is somewhat harder to handle. We provide a complete answer in the case of $m=3$ and arbitrary $n$, as well as some general results for arbitrary $m$. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case $n=3$, $m=2$. Finally, we provide explicit examples illustrating the various possibilities. Our initial motivation for considering this problem comes from the geometric study of hyperbolic conservative systems in one spatial dimension.

math.DG↗

arXiv2016

On convergence of exterior solutions to radial Cauchy solutions for $\square_{1+3}U=0$

Consider the Cauchy problem for the 3-d linear wave equation $\square_{1+3}U=0$ with radial initial data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. A standard result gives that $U$ belongs to $C([0,T];H^s(\mathbb{R}^3))$ whenever $(Φ,Ψ)\in H^s\times H^{s-1}(\mathbb{R}^3)$. In this note we are interested in the question of how $U$ can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls $B_\varepsilon$ about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of $H^s$ well-posedness for the wave equation. We show how explicit solution formulae yield convergence and optimal regularity for the Cauchy solution via exterior solutions, when the latter are extended continuously as constants on $B_\varepsilon$ at each time. We establish that for $s=2$ the solution $U$ can be realized as an $H^2$-limit (uniformly in time) of exterior solutions on $\mathbb{R}^3\setminus B_\varepsilon$ satisfying vanishing Neumann conditions along $|x|=\varepsilon$, as $\varepsilon\downarrow 0$. Similarly for $s=1$: $U$ is then an $H^1$-limit of exterior solutions satisfying vanishing Dirichlet conditions along $|x|=\varepsilon$.

math.AP↗

arXiv2015

Radial solutions to the Cauchy problem for $\square_{1+3}U=0$ as limits of exterior solutions

We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the 3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric Cauchy problem. In treating the 3-d wave equation we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3-d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in $C([0,T);H^s(\mathbb{R}^3))$ whenever $(Φ,Ψ)\in H^s\times H^{s-1}(\mathbb{R}^3)$. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in $H^2\times H^1(\mathbb{R}^3)$ we obtain $H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via exterior Dirichlet solutions.

math.AP↗

arXiv2012

TVD Fields and Isentropic Gas Flow

Little is known about global existence of large-variation solutions to Cauchy problems for systems of conservation laws in one space dimension. Besides results for $L^\infty$ data via compensated compactness, the existence of global BV solutions for arbitrary BV data remains an outstanding open problem. In particular, it is not known if isentropic gas dynamics admits an a priori variation bound which applies to all BV data. In a few cases such results are available: scalar equations, Temple class systems, $2\times 2$-systems satisfying Bakhvalov's condition, and, in particular, isothermal gas dynamics. In each of these cases the equations admit a TVD (Total Variation Diminishing) field: a scalar function defined on state space whose spatial variation along entropic solutions does not increase in time. In this paper we consider strictly hyperbolic $2\times 2$-systems and derive a representation result for scalar fields that are TVD across all pairwise wave interactions, when the latter are resolved as in the Glimm scheme. We then use this to show that isentropic gas dynamics with a $γ$-law pressure function does not admit any nontrivial TVD field of this type.

math.AP↗

arXiv2011

Pairwise wave interactions in ideal polytropic gases

We consider the problem of resolving all pairwise interactions of shock waves, contact waves, and rarefaction waves in 1-dimensional flow of an ideal polytropic gas. Resolving an interaction means here to determine the types of the three outgoing (backward, contact, and forward) waves in the Riemann problem defined by the extreme left and right states of the two incoming waves, together with possible vacuum formation. This problem has been considered by several authors and turns out to be surprisingly involved. For each type of interaction (head-on, involving a contact, or overtaking) the outcome depends on the strengths of the incoming waves. In the case of overtaking waves the type of the reflected wave also depends on the value of the adiabatic constant. Our analysis provides a complete breakdown and gives the exact outcome of each interaction.

math.AP↗